fact stringlengths 9 24k | type stringclasses 2
values | library stringclasses 5
values | imports listlengths 0 0 | filename stringclasses 5
values | symbolic_name stringlengths 1 24 | docstring stringlengths 12 292k ⌀ |
|---|---|---|---|---|---|---|
impcomd : |- ( ph -> ( ( ch /\ ps ) -> th ) ) | theorem | set | [] | set.mm | impcomd | Importation deduction with commuted antecedents. (Contributed by Peter Mazsa, 24-Sep-2022.) (Proof shortened by Wolf Lammen, 22-Oct-2022.) |
ex : |- ( ph -> ( ps -> ch ) ) | theorem | set | [] | set.mm | ex | Exportation inference. (This theorem used to be labeled "exp" but was changed to "ex" so as not to conflict with the math token "exp", per the June 2006 Metamath spec change.) A translation of natural deduction rule ` -> ` I ( ` -> ` introduction), see ~ natded . (Contributed by NM, 3-Jan-1993.) (Proof shortened by Eri... |
expcom : |- ( ps -> ( ph -> ch ) ) | theorem | set | [] | set.mm | expcom | Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |
expdcom : |- ( ps -> ( ch -> ( ph -> th ) ) ) | theorem | set | [] | set.mm | expdcom | Commuted form of ~ expd . (Contributed by Alan Sare, 18-Mar-2012.) Shorten ~ expd . (Revised by Wolf Lammen, 28-Jul-2022.) |
expd : |- ( ph -> ( ps -> ( ch -> th ) ) ) | theorem | set | [] | set.mm | expd | Exportation deduction. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 28-Jul-2022.) |
expcomd : |- ( ph -> ( ch -> ( ps -> th ) ) ) | theorem | set | [] | set.mm | expcomd | Deduction form of ~ expcom . (Contributed by Alan Sare, 22-Jul-2012.) |
imp31 : |- ( ( ( ph /\ ps ) /\ ch ) -> th ) | theorem | set | [] | set.mm | imp31 | An importation inference. (Contributed by NM, 26-Apr-1994.) |
imp32 : |- ( ( ph /\ ( ps /\ ch ) ) -> th ) | theorem | set | [] | set.mm | imp32 | An importation inference. (Contributed by NM, 26-Apr-1994.) |
exp31 : |- ( ph -> ( ps -> ( ch -> th ) ) ) | theorem | set | [] | set.mm | exp31 | An exportation inference. (Contributed by NM, 26-Apr-1994.) |
exp32 : |- ( ph -> ( ps -> ( ch -> th ) ) ) | theorem | set | [] | set.mm | exp32 | An exportation inference. (Contributed by NM, 26-Apr-1994.) |
imp4b : |- ( ( ph /\ ps ) -> ( ( ch /\ th ) -> ta ) ) | theorem | set | [] | set.mm | imp4b | An importation inference. (Contributed by NM, 26-Apr-1994.) Shorten ~ imp4a . (Revised by Wolf Lammen, 19-Jul-2021.) |
imp4a : |- ( ph -> ( ps -> ( ( ch /\ th ) -> ta ) ) ) | theorem | set | [] | set.mm | imp4a | An importation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Jul-2021.) |
imp4c : |- ( ph -> ( ( ( ps /\ ch ) /\ th ) -> ta ) ) | theorem | set | [] | set.mm | imp4c | An importation inference. (Contributed by NM, 26-Apr-1994.) |
imp4d : |- ( ph -> ( ( ps /\ ( ch /\ th ) ) -> ta ) ) | theorem | set | [] | set.mm | imp4d | An importation inference. (Contributed by NM, 26-Apr-1994.) |
imp41 : |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) | theorem | set | [] | set.mm | imp41 | An importation inference. (Contributed by NM, 26-Apr-1994.) |
imp42 : |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta ) | theorem | set | [] | set.mm | imp42 | An importation inference. (Contributed by NM, 26-Apr-1994.) |
imp43 : |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) | theorem | set | [] | set.mm | imp43 | An importation inference. (Contributed by NM, 26-Apr-1994.) |
imp44 : |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ta ) | theorem | set | [] | set.mm | imp44 | An importation inference. (Contributed by NM, 26-Apr-1994.) |
imp45 : |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ta ) | theorem | set | [] | set.mm | imp45 | An importation inference. (Contributed by NM, 26-Apr-1994.) |
exp4b : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) | theorem | set | [] | set.mm | exp4b | An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) Shorten ~ exp4a . (Revised by Wolf Lammen, 20-Jul-2021.) |
exp4a : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) | theorem | set | [] | set.mm | exp4a | An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2021.) |
exp4c : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) | theorem | set | [] | set.mm | exp4c | An exportation inference. (Contributed by NM, 26-Apr-1994.) |
exp4d : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) | theorem | set | [] | set.mm | exp4d | An exportation inference. (Contributed by NM, 26-Apr-1994.) |
exp41 : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) | theorem | set | [] | set.mm | exp41 | An exportation inference. (Contributed by NM, 26-Apr-1994.) |
exp42 : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) | theorem | set | [] | set.mm | exp42 | An exportation inference. (Contributed by NM, 26-Apr-1994.) |
exp43 : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) | theorem | set | [] | set.mm | exp43 | An exportation inference. (Contributed by NM, 26-Apr-1994.) |
exp44 : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) | theorem | set | [] | set.mm | exp44 | An exportation inference. (Contributed by NM, 26-Apr-1994.) |
exp45 : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) | theorem | set | [] | set.mm | exp45 | An exportation inference. (Contributed by NM, 26-Apr-1994.) |
imp5d : |- ( ( ( ph /\ ps ) /\ ch ) -> ( ( th /\ ta ) -> et ) ) | theorem | set | [] | set.mm | imp5d | An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
imp5a : |- ( ph -> ( ps -> ( ch -> ( ( th /\ ta ) -> et ) ) ) ) | theorem | set | [] | set.mm | imp5a | An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof shortened by Wolf Lammen, 2-Aug-2022.) |
imp5g : |- ( ( ph /\ ps ) -> ( ( ( ch /\ th ) /\ ta ) -> et ) ) | theorem | set | [] | set.mm | imp5g | An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
imp55 : |- ( ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) /\ ta ) -> et ) | theorem | set | [] | set.mm | imp55 | An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
imp511 : |- ( ( ph /\ ( ( ps /\ ( ch /\ th ) ) /\ ta ) ) -> et ) | theorem | set | [] | set.mm | imp511 | An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
exp5c : |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) | theorem | set | [] | set.mm | exp5c | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
exp5j : |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) | theorem | set | [] | set.mm | exp5j | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
exp5l : |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) | theorem | set | [] | set.mm | exp5l | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
exp53 : |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) | theorem | set | [] | set.mm | exp53 | An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.) |
pm3.3 : |- ( ( ( ph /\ ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) ) | theorem | set | [] | set.mm | pm3.3 | Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) |
pm3.31 : |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph /\ ps ) -> ch ) ) | theorem | set | [] | set.mm | pm3.31 | Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) |
impexp : |- ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) | theorem | set | [] | set.mm | impexp | Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) |
impancom : |- ( ( ph /\ ch ) -> ( ps -> th ) ) | theorem | set | [] | set.mm | impancom | Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.) |
expdimp : |- ( ( ph /\ ps ) -> ( ch -> th ) ) | theorem | set | [] | set.mm | expdimp | A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.) |
expimpd : |- ( ph -> ( ( ps /\ ch ) -> th ) ) | theorem | set | [] | set.mm | expimpd | Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.) |
impr : |- ( ( ph /\ ( ps /\ ch ) ) -> th ) | theorem | set | [] | set.mm | impr | Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.) |
impl : |- ( ( ( ph /\ ps ) /\ ch ) -> th ) | theorem | set | [] | set.mm | impl | Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.) |
expr : |- ( ( ph /\ ps ) -> ( ch -> th ) ) | theorem | set | [] | set.mm | expr | Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.) |
expl : |- ( ph -> ( ( ps /\ ch ) -> th ) ) | theorem | set | [] | set.mm | expl | Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.) |
ancoms : |- ( ( ps /\ ph ) -> ch ) | theorem | set | [] | set.mm | ancoms | Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.) |
pm3.22 : |- ( ( ph /\ ps ) -> ( ps /\ ph ) ) | theorem | set | [] | set.mm | pm3.22 | Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) |
ancom : |- ( ( ph /\ ps ) <-> ( ps /\ ph ) ) | theorem | set | [] | set.mm | ancom | Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Wolf Lammen, 4-Nov-2012.) |
ancomd : |- ( ph -> ( ch /\ ps ) ) | theorem | set | [] | set.mm | ancomd | Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.) |
biancomi : |- ( ph <-> ( ps /\ ch ) ) | theorem | set | [] | set.mm | biancomi | Commuting conjunction in a biconditional. (Contributed by Peter Mazsa, 17-Jun-2018.) |
biancomd : |- ( ph -> ( ps <-> ( ch /\ th ) ) ) | theorem | set | [] | set.mm | biancomd | Commuting conjunction in a biconditional, deduction form. (Contributed by Peter Mazsa, 3-Oct-2018.) |
ancomst : |- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) ) | theorem | set | [] | set.mm | ancomst | Closed form of ~ ancoms . (Contributed by Alan Sare, 31-Dec-2011.) |
ancomsd : |- ( ph -> ( ( ch /\ ps ) -> th ) ) | theorem | set | [] | set.mm | ancomsd | Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.) |
anasss : |- ( ( ph /\ ( ps /\ ch ) ) -> th ) | theorem | set | [] | set.mm | anasss | Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.) |
anassrs : |- ( ( ( ph /\ ps ) /\ ch ) -> th ) | theorem | set | [] | set.mm | anassrs | Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.) |
anass : |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) ) | theorem | set | [] | set.mm | anass | Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) |
pm3.2 : |- ( ph -> ( ps -> ( ph /\ ps ) ) ) | theorem | set | [] | set.mm | pm3.2 | Join antecedents with conjunction ("conjunction introduction"). Theorem *3.2 of [WhiteheadRussell] p. 111. Its associated inference is ~ pm3.2i and its associated deduction is ~ jca (and the double deduction is ~ jcad ). See ~ pm3.2im for a version using only implication and negation. (Contributed by NM, 5-Jan-1993.) (... |
pm3.2i : |- ( ph /\ ps ) | theorem | set | [] | set.mm | pm3.2i | Infer conjunction of premises. Inference associated with ~ pm3.2 . Its associated deduction is ~ jca (and the double deduction is ~ jcad ). (Contributed by NM, 21-Jun-1993.) |
pm3.21 : |- ( ph -> ( ps -> ( ps /\ ph ) ) ) | theorem | set | [] | set.mm | pm3.21 | Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.) |
pm3.43i : |- ( ( ph -> ps ) -> ( ( ph -> ch ) -> ( ph -> ( ps /\ ch ) ) ) ) | theorem | set | [] | set.mm | pm3.43i | Nested conjunction of antecedents. (Contributed by NM, 4-Jan-1993.) |
pm3.43 : |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) -> ( ph -> ( ps /\ ch ) ) ) | theorem | set | [] | set.mm | pm3.43 | Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) |
dfbi2 : |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) | theorem | set | [] | set.mm | dfbi2 | A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 24-Jan-1993.) |
dfbi : |- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) ) | theorem | set | [] | set.mm | dfbi | Definition ~ df-bi rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional. (Contributed by NM, 15-Aug-2008.) |
biimpa : |- ( ( ph /\ ps ) -> ch ) | theorem | set | [] | set.mm | biimpa | Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.) |
biimpar : |- ( ( ph /\ ch ) -> ps ) | theorem | set | [] | set.mm | biimpar | Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.) |
biimpac : |- ( ( ps /\ ph ) -> ch ) | theorem | set | [] | set.mm | biimpac | Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.) |
biimparc : |- ( ( ch /\ ph ) -> ps ) | theorem | set | [] | set.mm | biimparc | Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.) |
adantr : |- ( ( ph /\ ch ) -> ps ) | theorem | set | [] | set.mm | adantr | Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.) |
adantl : |- ( ( ch /\ ph ) -> ps ) | theorem | set | [] | set.mm | adantl | Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) |
simpl : |- ( ( ph /\ ps ) -> ph ) | theorem | set | [] | set.mm | simpl | Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 14-Jun-2022.) |
simpli : |- ph | theorem | set | [] | set.mm | simpli | Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) |
simpr : |- ( ( ph /\ ps ) -> ps ) | theorem | set | [] | set.mm | simpr | Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 14-Jun-2022.) |
simpri : |- ps | theorem | set | [] | set.mm | simpri | Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) |
intnan : |- -. ( ps /\ ph ) | theorem | set | [] | set.mm | intnan | Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.) |
intnanr : |- -. ( ph /\ ps ) | theorem | set | [] | set.mm | intnanr | Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.) |
intnand : |- ( ph -> -. ( ch /\ ps ) ) | theorem | set | [] | set.mm | intnand | Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.) |
intnanrd : |- ( ph -> -. ( ps /\ ch ) ) | theorem | set | [] | set.mm | intnanrd | Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.) |
adantld : |- ( ph -> ( ( th /\ ps ) -> ch ) ) | theorem | set | [] | set.mm | adantld | Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.) |
adantrd : |- ( ph -> ( ( ps /\ th ) -> ch ) ) | theorem | set | [] | set.mm | adantrd | Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.) |
pm3.41 : |- ( ( ph -> ch ) -> ( ( ph /\ ps ) -> ch ) ) | theorem | set | [] | set.mm | pm3.41 | Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) |
pm3.42 : |- ( ( ps -> ch ) -> ( ( ph /\ ps ) -> ch ) ) | theorem | set | [] | set.mm | pm3.42 | Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) |
simpld : |- ( ph -> ps ) | theorem | set | [] | set.mm | simpld | Deduction eliminating a conjunct. A translation of natural deduction rule ` /\ ` EL ( ` /\ ` elimination left), see ~ natded . (Contributed by NM, 26-May-1993.) |
simprd : |- ( ph -> ch ) | theorem | set | [] | set.mm | simprd | Deduction eliminating a conjunct. (Contributed by NM, 14-May-1993.) A translation of natural deduction rule ` /\ ` ER ( ` /\ ` elimination right), see ~ natded . (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
simprbi : |- ( ph -> ch ) | theorem | set | [] | set.mm | simprbi | Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.) |
simplbi : |- ( ph -> ps ) | theorem | set | [] | set.mm | simplbi | Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.) |
simprbda : |- ( ( ph /\ ps ) -> ch ) | theorem | set | [] | set.mm | simprbda | Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) |
simplbda : |- ( ( ph /\ ps ) -> th ) | theorem | set | [] | set.mm | simplbda | Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) |
simplbi2 : |- ( ps -> ( ch -> ph ) ) | theorem | set | [] | set.mm | simplbi2 | Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.) |
simplbi2comt : |- ( ( ph <-> ( ps /\ ch ) ) -> ( ch -> ( ps -> ph ) ) ) | theorem | set | [] | set.mm | simplbi2comt | Closed form of ~ simplbi2com . (Contributed by Alan Sare, 22-Jul-2012.) |
simplbi2com : |- ( ch -> ( ps -> ph ) ) | theorem | set | [] | set.mm | simplbi2com | A deduction eliminating a conjunct, similar to ~ simplbi2 . (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.) |
simpl2im : |- ( ph -> th ) | theorem | set | [] | set.mm | simpl2im | Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.) (Proof shortened by Wolf Lammen, 26-Mar-2022.) |
simplbiim : |- ( ph -> th ) | theorem | set | [] | set.mm | simplbiim | Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 26-Mar-2022.) |
impel : |- ( ( ph /\ th ) -> ch ) | theorem | set | [] | set.mm | impel | An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.) |
mpan9 : |- ( ( ph /\ ch ) -> th ) | theorem | set | [] | set.mm | mpan9 | Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
sylan9 : |- ( ( ph /\ th ) -> ( ps -> ta ) ) | theorem | set | [] | set.mm | sylan9 | Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
sylan9r : |- ( ( th /\ ph ) -> ( ps -> ta ) ) | theorem | set | [] | set.mm | sylan9r | Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.) |
sylan9bb : |- ( ( ph /\ th ) -> ( ps <-> ta ) ) | theorem | set | [] | set.mm | sylan9bb | Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.) |
sylan9bbr : |- ( ( th /\ ph ) -> ( ps <-> ta ) ) | theorem | set | [] | set.mm | sylan9bbr | Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.) |
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