P. 12 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1101-1200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	simp3l1 1101	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ph )
Theorem	simp3l2 1102	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ps )
Theorem	simp3l3 1103	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ch )
Theorem	simp3r1 1104	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ph )
Theorem	simp3r2 1105	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ps )
Theorem	simp3r3 1106	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ch )
Theorem	simp11l 1107	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta /\ et ) -> ph )
Theorem	simp11r 1108	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta /\ et ) -> ps )
Theorem	simp12l 1109	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta /\ et ) -> ph )
Theorem	simp12r 1110	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta /\ et ) -> ps )
Theorem	simp13l 1111	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta /\ et ) -> ph )
Theorem	simp13r 1112	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta /\ et ) -> ps )
Theorem	simp21l 1113	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) /\ et ) -> ph )
Theorem	simp21r 1114	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) /\ et ) -> ps )
Theorem	simp22l 1115	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) /\ et ) -> ph )
Theorem	simp22r 1116	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) /\ et ) -> ps )
Theorem	simp23l 1117	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) /\ et ) -> ph )
Theorem	simp23r 1118	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) /\ et ) -> ps )
Theorem	simp31l 1119	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ph )
Theorem	simp31r 1120	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ps )
Theorem	simp32l 1121	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ph )
Theorem	simp32r 1122	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ps )
Theorem	simp33l 1123	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ph )
Theorem	simp33r 1124	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ et /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ps )
Theorem	simp111 1125	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ph )
Theorem	simp112 1126	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ps )
Theorem	simp113 1127	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ch )
Theorem	simp121 1128	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ph )
Theorem	simp122 1129	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ps )
Theorem	simp123 1130	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ch )
Theorem	simp131 1131	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ph )
Theorem	simp132 1132	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ps )
Theorem	simp133 1133	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ch )
Theorem	simp211 1134	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ph )
Theorem	simp212 1135	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ps )
Theorem	simp213 1136	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ch )
Theorem	simp221 1137	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ph )
Theorem	simp222 1138	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ps )
Theorem	simp223 1139	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ch )
Theorem	simp231 1140	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ph )
Theorem	simp232 1141	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ps )
Theorem	simp233 1142	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ch )
Theorem	simp311 1143	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ph )
Theorem	simp312 1144	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ps )
Theorem	simp313 1145	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ch )
Theorem	simp321 1146	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ph )
Theorem	simp322 1147	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ps )
Theorem	simp323 1148	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ch )
Theorem	simp331 1149	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ph )
Theorem	simp332 1150	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ps )
Theorem	simp333 1151	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ch )
Theorem	3adantl1 1152	Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ( ta /\ ph /\ ps ) /\ ch ) -> th )
Theorem	3adantl2 1153	Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ( ph /\ ta /\ ps ) /\ ch ) -> th )
Theorem	3adantl3 1154	Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ( ph /\ ps /\ ta ) /\ ch ) -> th )
Theorem	3adantr1 1155	Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ( ta /\ ps /\ ch ) ) -> th )
Theorem	3adantr2 1156	Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th )
Theorem	3adantr3 1157	Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th )
Theorem	3ad2antl1 1158	Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
|- ( ( ph /\ ch ) -> th )   =>    |- ( ( ( ph /\ ps /\ ta ) /\ ch ) -> th )
Theorem	3ad2antl2 1159	Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
|- ( ( ph /\ ch ) -> th )   =>    |- ( ( ( ps /\ ph /\ ta ) /\ ch ) -> th )
Theorem	3ad2antl3 1160	Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
|- ( ( ph /\ ch ) -> th )   =>    |- ( ( ( ps /\ ta /\ ph ) /\ ch ) -> th )
Theorem	3ad2antr1 1161	Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
|- ( ( ph /\ ch ) -> th )   =>    |- ( ( ph /\ ( ch /\ ps /\ ta ) ) -> th )
Theorem	3ad2antr2 1162	Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
|- ( ( ph /\ ch ) -> th )   =>    |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th )
Theorem	3ad2antr3 1163	Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
|- ( ( ph /\ ch ) -> th )   =>    |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th )
Theorem	3anibar 1164	Remove a hypothesis from the second member of a biimplication. (Contributed by FL, 22-Jul-2008.)
|- ( ( ph /\ ps /\ ch ) -> ( th <-> ( ch /\ ta ) ) )   =>    |- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) )
Theorem	3mix1 1165	Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
|- ( ph -> ( ph \/ ps \/ ch ) )
Theorem	3mix2 1166	Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
|- ( ph -> ( ps \/ ph \/ ch ) )
Theorem	3mix3 1167	Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
|- ( ph -> ( ps \/ ch \/ ph ) )
Theorem	3mix1i 1168	Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
|- ph   =>    |- ( ph \/ ps \/ ch )
Theorem	3mix2i 1169	Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
|- ph   =>    |- ( ps \/ ph \/ ch )
Theorem	3mix3i 1170	Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
|- ph   =>    |- ( ps \/ ch \/ ph )
Theorem	3mix1d 1171	Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
|- ( ph -> ps )   =>    |- ( ph -> ( ps \/ ch \/ th ) )
Theorem	3mix2d 1172	Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
|- ( ph -> ps )   =>    |- ( ph -> ( ch \/ ps \/ th ) )
Theorem	3mix3d 1173	Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
|- ( ph -> ps )   =>    |- ( ph -> ( ch \/ th \/ ps ) )
Theorem	3pm3.2i 1174	Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
|- ph   &    |- ps   &    |- ch   =>    |- ( ph /\ ps /\ ch )
Theorem	pm3.2an3 1175	Version of pm3.2 447 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
|- ( ph -> ( ps -> ( ch -> ( ph /\ ps /\ ch ) ) ) )
Theorem	3jca 1176	Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( ph -> th )   =>    |- ( ph -> ( ps /\ ch /\ th ) )
Theorem	3jcad 1177	Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( ps -> th ) )   &    |- ( ph -> ( ps -> ta ) )   =>    |- ( ph -> ( ps -> ( ch /\ th /\ ta ) ) )
Theorem	mpbir3an 1178	Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.)
|- ps   &    |- ch   &    |- th   &    |- ( ph <-> ( ps /\ ch /\ th ) )   =>    |- ph
Theorem	mpbir3and 1179	Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)
|- ( ph -> ch )   &    |- ( ph -> th )   &    |- ( ph -> ta )   &    |- ( ph -> ( ps <-> ( ch /\ th /\ ta ) ) )   =>    |- ( ph -> ps )
Theorem	syl3anbrc 1180	Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( ph -> th )   &    |- ( ta <-> ( ps /\ ch /\ th ) )   =>    |- ( ph -> ta )
Theorem	3anim123i 1181	Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
|- ( ph -> ps )   &    |- ( ch -> th )   &    |- ( ta -> et )   =>    |- ( ( ph /\ ch /\ ta ) -> ( ps /\ th /\ et ) )
Theorem	3anim1i 1182	Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
|- ( ph -> ps )   =>    |- ( ( ph /\ ch /\ th ) -> ( ps /\ ch /\ th ) )
Theorem	3anim2i 1183	Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
|- ( ph -> ps )   =>    |- ( ( ch /\ ph /\ th ) -> ( ch /\ ps /\ th ) )
Theorem	3anim3i 1184	Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
|- ( ph -> ps )   =>    |- ( ( ch /\ th /\ ph ) -> ( ch /\ th /\ ps ) )
Theorem	3anbi123i 1185	Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   &    |- ( ta <-> et )   =>    |- ( ( ph /\ ch /\ ta ) <-> ( ps /\ th /\ et ) )
Theorem	3orbi123i 1186	Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   &    |- ( ta <-> et )   =>    |- ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) )
Theorem	3anbi1i 1187	Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
|- ( ph <-> ps )   =>    |- ( ( ph /\ ch /\ th ) <-> ( ps /\ ch /\ th ) )
Theorem	3anbi2i 1188	Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
|- ( ph <-> ps )   =>    |- ( ( ch /\ ph /\ th ) <-> ( ch /\ ps /\ th ) )
Theorem	3anbi3i 1189	Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
|- ( ph <-> ps )   =>    |- ( ( ch /\ th /\ ph ) <-> ( ch /\ th /\ ps ) )
Theorem	3imp 1190	Importation inference. (Contributed by NM, 8-Apr-1994.)
|- ( ph -> ( ps -> ( ch -> th ) ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	3impa 1191	Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	3impb 1192	Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	3impia 1193	Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
|- ( ( ph /\ ps ) -> ( ch -> th ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	3impib 1194	Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	3exp 1195	Exportation inference. (Contributed by NM, 30-May-1994.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ph -> ( ps -> ( ch -> th ) ) )
Theorem	3expa 1196	Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ph /\ ps ) /\ ch ) -> th )
Theorem	3expb 1197	Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ph /\ ( ps /\ ch ) ) -> th )
Theorem	3expia 1198	Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ph /\ ps ) -> ( ch -> th ) )
Theorem	3expib 1199	Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ph -> ( ( ps /\ ch ) -> th ) )
Theorem	3com12 1200	Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ps /\ ph /\ ch ) -> th )