mmtheorems12 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1101-1200 - Page 12 of 175

Type	Label	Description
Statement
Theorem	syl22anc 1101	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (((ps /\ ch) /\ (th /\ ta)) -> et)   =>   |- (ph -> et)
Theorem	syl13anc 1102	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- ((ps /\ (ch /\ th /\ ta)) -> et)   =>   |- (ph -> et)
Theorem	syl31anc 1103	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (((ps /\ ch /\ th) /\ ta) -> et)   =>   |- (ph -> et)
Theorem	syl112anc 1104	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- ((ps /\ ch /\ (th /\ ta)) -> et)   =>   |- (ph -> et)
Theorem	syl121anc 1105	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- ((ps /\ (ch /\ th) /\ ta) -> et)   =>   |- (ph -> et)
Theorem	syl211anc 1106	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (((ps /\ ch) /\ th /\ ta) -> et)   =>   |- (ph -> et)
Theorem	syl23anc 1107	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (((ps /\ ch) /\ (th /\ ta /\ et)) -> ze)   =>   |- (ph -> ze)
Theorem	syl32anc 1108	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et)) -> ze)   =>   |- (ph -> ze)
Theorem	syl122anc 1109	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- ((ps /\ (ch /\ th) /\ (ta /\ et)) -> ze)   =>   |- (ph -> ze)
Theorem	syl212anc 1110	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (((ps /\ ch) /\ th /\ (ta /\ et)) -> ze)   =>   |- (ph -> ze)
Theorem	syl221anc 1111	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (((ps /\ ch) /\ (th /\ ta) /\ et) -> ze)   =>   |- (ph -> ze)
Theorem	syl113anc 1112	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- ((ps /\ ch /\ (th /\ ta /\ et)) -> ze)   =>   |- (ph -> ze)
Theorem	syl131anc 1113	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- ((ps /\ (ch /\ th /\ ta) /\ et) -> ze)   =>   |- (ph -> ze)
Theorem	syl311anc 1114	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (((ps /\ ch /\ th) /\ ta /\ et) -> ze)   =>   |- (ph -> ze)
Theorem	syl33anc 1115	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et /\ ze)) -> si)   =>   |- (ph -> si)
Theorem	syl222anc 1116	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (((ps /\ ch) /\ (th /\ ta) /\ (et /\ ze)) -> si)   =>   |- (ph -> si)
Theorem	syl123anc 1117	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- ((ps /\ (ch /\ th) /\ (ta /\ et /\ ze)) -> si)   =>   |- (ph -> si)
Theorem	syl213anc 1118	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (((ps /\ ch) /\ th /\ (ta /\ et /\ ze)) -> si)   =>   |- (ph -> si)
Theorem	syl231anc 1119	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (((ps /\ ch) /\ (th /\ ta /\ et) /\ ze) -> si)   =>   |- (ph -> si)
Theorem	syl133anc 1120	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- ((ps /\ (ch /\ th /\ ta) /\ (et /\ ze /\ si)) -> rh)   =>   |- (ph -> rh)
Theorem	syl313anc 1121	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (((ps /\ ch /\ th) /\ ta /\ (et /\ ze /\ si)) -> rh)   =>   |- (ph -> rh)
Theorem	syl331anc 1122	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et /\ ze) /\ si) -> rh)   =>   |- (ph -> rh)
Theorem	syl223anc 1123	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (((ps /\ ch) /\ (th /\ ta) /\ (et /\ ze /\ si)) -> rh)   =>   |- (ph -> rh)
Theorem	syl232anc 1124	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (((ps /\ ch) /\ (th /\ ta /\ et) /\ (ze /\ si)) -> rh)   =>   |- (ph -> rh)
Theorem	syl322anc 1125	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et) /\ (ze /\ si)) -> rh)   =>   |- (ph -> rh)
Theorem	syl233anc 1126	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (ph -> rh)   &   |- (((ps /\ ch) /\ (th /\ ta /\ et) /\ (ze /\ si /\ rh)) -> mu)   =>   |- (ph -> mu)
Theorem	syl323anc 1127	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (ph -> rh)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et) /\ (ze /\ si /\ rh)) -> mu)   =>   |- (ph -> mu)
Theorem	syl332anc 1128	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (ph -> rh)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et /\ ze) /\ (si /\ rh)) -> mu)   =>   |- (ph -> mu)
Theorem	syl333anc 1129	A syllogism inference combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (ph -> rh)   &   |- (ph -> mu)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et /\ ze) /\ (si /\ rh /\ mu)) -> la)   =>   |- (ph -> la)
Theorem	syl3an1 1130	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ph)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syl3an2 1131	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ps)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3an3 1132	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ch)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syl3an1b 1133	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ph)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syl3an2b 1134	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ps)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3an3b 1135	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ch)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syl3an1br 1136	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ph <-> ta)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syl3an2br 1137	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ps <-> ta)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3an3br 1138	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ch <-> ta)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syl3an 1139	A triple syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ph)   &   |- (et -> ps)   &   |- (ze -> ch)   =>   |- ((ta /\ et /\ ze) -> th)
Theorem	syl3anb 1140	A triple syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ph)   &   |- (et <-> ps)   &   |- (ze <-> ch)   =>   |- ((ta /\ et /\ ze) -> th)
Theorem	syl3anbr 1141	A triple syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ph <-> ta)   &   |- (ps <-> et)   &   |- (ch <-> ze)   =>   |- ((ta /\ et /\ ze) -> th)
Theorem	syld3an3 1142	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- ((ph /\ ps /\ ta) -> ch)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syld3an1 1143	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- ((ta /\ ps /\ ch) -> ph)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syld3an2 1144	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- ((ph /\ ta /\ ch) -> ps)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3anl1 1145	A syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ph)   =>   |- (((et /\ ps /\ ch) /\ th) -> ta)
Theorem	syl3anl2 1146	A syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ps)   =>   |- (((ph /\ et /\ ch) /\ th) -> ta)
Theorem	syl3anl3 1147	A syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ch)   =>   |- (((ph /\ ps /\ et) /\ th) -> ta)
Theorem	syl3anl 1148	A triple syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ph)   &   |- (ze -> ps)   &   |- (si -> ch)   =>   |- (((et /\ ze /\ si) /\ th) -> ta)
Theorem	syl3anr1 1149	A syllogism inference.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   &   |- (et -> ps)   =>   |- ((ph /\ (et /\ ch /\ th)) -> ta)
Theorem	syl3anr2 1150	A syllogism inference.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   &   |- (et -> ch)   =>   |- ((ph /\ (ps /\ et /\ th)) -> ta)
Theorem	syl3anr3 1151	A syllogism inference.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   &   |- (et -> th)   =>   |- ((ph /\ (ps /\ ch /\ et)) -> ta)
Theorem	3impdi 1152	Importation inference (undistribute conjunction).
|- (((ph /\ ps) /\ (ph /\ ch)) -> th)   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impdir 1153	Importation inference (undistribute conjunction).
|- (((ph /\ ps) /\ (ch /\ ps)) -> th)   =>   |- ((ph /\ ch /\ ps) -> th)
Theorem	3anidm12 1154	Inference from idempotent law for conjunction.
|- ((ph /\ ph /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	3anidm13 1155	Inference from idempotent law for conjunction.
|- ((ph /\ ps /\ ph) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	3anidm23 1156	Inference from idempotent law for conjunction.
|- ((ph /\ ps /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	3ori 1157	Infer implication from triple disjunction.
|- (ph \/ ps \/ ch)   =>   |- ((-. ph /\ -. ps) -> ch)
Theorem	3jao 1158	Disjunction of 3 antecedents.
|- (((ph -> ps) /\ (ch -> ps) /\ (th -> ps)) -> ((ph \/ ch \/ th) -> ps))
Theorem	3jaob 1159	Disjunction of 3 antecedents.
|- (((ph \/ ch \/ th) -> ps) <-> ((ph -> ps) /\ (ch -> ps) /\ (th -> ps)))
Theorem	3jaoi 1160	Disjunction of 3 antecedents (inference).
|- (ph -> ps)   &   |- (ch -> ps)   &   |- (th -> ps)   =>   |- ((ph \/ ch \/ th) -> ps)
Theorem	3jaod 1161	Disjunction of 3 antecedents (deduction).
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ch))   &   |- (ph -> (ta -> ch))   =>   |- (ph -> ((ps \/ th \/ ta) -> ch))
Theorem	3jaoian 1162	Disjunction of 3 antecedents (inference).
|- ((ph /\ ps) -> ch)   &   |- ((th /\ ps) -> ch)   &   |- ((ta /\ ps) -> ch)   =>   |- (((ph \/ th \/ ta) /\ ps) -> ch)
Theorem	3jaodan 1163	Disjunction of 3 antecedents (deduction).
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ th) -> ch)   &   |- ((ph /\ ta) -> ch)   =>   |- ((ph /\ (ps \/ th \/ ta)) -> ch)
Theorem	3jaao 1164	Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (The proof was shortened by Andrew Salmon, 13-May-2011.)
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   &   |- (et -> (ze -> ch))   =>   |- ((ph /\ th /\ et) -> ((ps \/ ta \/ ze) -> ch))
Theorem	3jaaoOLD 1165	Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.)
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   &   |- (et -> (ze -> ch))   =>   |- ((ph /\ th /\ et) -> ((ps \/ ta \/ ze) -> ch))
Theorem	syl3an9b 1166	Nested syllogism inference conjoining 3 dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta))   &   |- (et -> (ta <-> ze))   =>   |- ((ph /\ th /\ et) -> (ps <-> ze))
Theorem	3orbi123d 1167	Deduction joining 3 equivalences to form equivalence of disjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   &   |- (ph -> (et <-> ze))   =>   |- (ph -> ((ps \/ th \/ et) <-> (ch \/ ta \/ ze)))
Theorem	3anbi123d 1168	Deduction joining 3 equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   &   |- (ph -> (et <-> ze))   =>   |- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ ze)))
Theorem	3anbi12d 1169	Deduction conjoining and adding a conjunct to equivalences.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ et)))
Theorem	3anbi13d 1170	Deduction conjoining and adding a conjunct to equivalences.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps /\ et /\ th) <-> (ch /\ et /\ ta)))
Theorem	3anbi23d 1171	Deduction conjoining and adding a conjunct to equivalences.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((et /\ ps /\ th) <-> (et /\ ch /\ ta)))
Theorem	3anbi1d 1172	Deduction adding conjuncts to an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps /\ th /\ ta) <-> (ch /\ th /\ ta)))
Theorem	3anbi2d 1173	Deduction adding conjuncts to an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th /\ ps /\ ta) <-> (th /\ ch /\ ta)))
Theorem	3anbi3d 1174	Deduction adding conjuncts to an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th /\ ta /\ ps) <-> (th /\ ta /\ ch)))
Theorem	3anim123d 1175	Deduction joining 3 implications to form implication of conjunctions.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta))   &   |- (ph -> (et -> ze))   =>   |- (ph -> ((ps /\ th /\ et) -> (ch /\ ta /\ ze)))
Theorem	3orim123d 1176	Deduction joining 3 implications to form implication of disjunctions.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta))   &   |- (ph -> (et -> ze))   =>   |- (ph -> ((ps \/ th \/ et) -> (ch \/ ta \/ ze)))
Theorem	an6 1177	Rearrangement of 6 conjuncts.
|- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> ((ph /\ th) /\ (ps /\ ta) /\ (ch /\ et)))
Theorem	mp3an1 1178	An inference based on modus ponens.
|- ph   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ch) -> th)
Theorem	mp3an2 1179	An inference based on modus ponens.
|- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mp3an3 1180	An inference based on modus ponens.
|- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mp3an12 1181	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ch -> th)
Theorem	mp3an13 1182	An inference based on modus ponens.
|- ph   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ps -> th)
Theorem	mp3an23 1183	An inference based on modus ponens.
|- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	mp3an1i 1184	An inference based on modus ponens.
|- ps   &   |- (ph -> ((ps /\ ch /\ th) -> ta))   =>   |- (ph -> ((ch /\ th) -> ta))
Theorem	mp3anl1 1185	An inference based on modus ponens.
|- ph   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ps /\ ch) /\ th) -> ta)
Theorem	mp3anl2 1186	An inference based on modus ponens.
|- ps   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ch) /\ th) -> ta)
Theorem	mp3anl3 1187	An inference based on modus ponens.
|- ch   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ps) /\ th) -> ta)
Theorem	mp3anr1 1188	An inference based on modus ponens.
|- ps   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ch /\ th)) -> ta)
Theorem	mp3anr2 1189	An inference based on modus ponens.
|- ch   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ th)) -> ta)
Theorem	mp3anr3 1190	An inference based on modus ponens.
|- th   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch)) -> ta)
Theorem	mp3an 1191	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th
Theorem	mpd3an3 1192	An inference based on modus ponens.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpd3an23 1193	An inference based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	biimp3a 1194	Infer implication from a logical equivalence. Similar to biimpa 460.
|- ((ph /\ ps) -> (ch <-> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	biimp3ar 1195	Infer implication from a logical equivalence. Similar to biimpar 461.
|- ((ph /\ ps) -> (ch <-> th))   =>   |- ((ph /\ ps /\ th) -> ch)
Theorem	3anandis 1196	Inference that undistributes a triple conjunction in the antecedent.
|- (((ph /\ ps) /\ (ph /\ ch) /\ (ph /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch /\ th)) -> ta)
Theorem	3anandirs 1197	Inference that undistributes a triple conjunction in the antecedent.
|- (((ph /\ th) /\ (ps /\ th) /\ (ch /\ th)) -> ta)   =>   |- (((ph /\ ps /\ ch) /\ th) -> ta)
Theorem	ecase23d 1198	Deduction for elimination by cases.
|- (ph -> -. ch)   &   |- (ph -> -. th)   &   |- (ph -> (ps \/ ch \/ th))   =>   |- (ph -> ps)
Theorem	3ecase 1199	Inference for elimination by cases.
|- (-. ph -> th)   &   |- (-. ps -> th)   &   |- (-. ch -> th)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th
Other axiomatizations of classical propositional calculus
Derive the Lukasiewicz axioms from Meredith's sole axiom
Theorem	meredith 1200	Carew Meredith's sole axiom for propositional calculus. This amazing formula is thought to be the shortest possible single axiom for propositional calculus with inference rule ax-mp 7, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 4, ax-2 5, and ax-3 6. Then from it we derive the Lukasiewicz axioms luk-1 1215, luk-2 1216, and luk-3 1217. Using these we finally re-derive our axioms as ax1 1226, ax2 1227, and ax3 1228, thus proving the equivalence of all three systems. C. A. Meredith, "Single Axioms for the Systems (C,N), (C,O) and (A,N) of the Two-Valued Propositional Calculus," The Journal of Computing Systems vol. 1 (1953), pp. 155-164. Meredith claimed to be close to a proof that this axiom is the shortest possible, but the proof was apparently never completed. An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic somewhat late in life after attending talks by Lukasiewicz and produced many remarkable results such as this axiom. From his obituary: "He did logic whenever time and opportunity presented themselves, and he did it on whatever materials came to hand: in a pub, his favored pint of porter within reach, he would use the inside of cigarette packs to write proofs for logical colleagues." (The proof was shortened by Andrew Salmon, 25-Jul-2011.)
|- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((ta -> ph) -> (th -> ph)))