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Theorem List for Metamath Proof Explorer - 1101-1200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimp221 1101 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta )  /\  ze )  ->  ph )
 
Theoremsimp222 1102 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta )  /\  ze )  ->  ps )
 
Theoremsimp223 1103 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta )  /\  ze )  ->  ch )
 
Theoremsimp231 1104 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) )  /\  ze )  ->  ph )
 
Theoremsimp232 1105 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) )  /\  ze )  ->  ps )
 
Theoremsimp233 1106 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) )  /\  ze )  ->  ch )
 
Theoremsimp311 1107 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )
 )  ->  ph )
 
Theoremsimp312 1108 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )
 )  ->  ps )
 
Theoremsimp313 1109 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )
 )  ->  ch )
 
Theoremsimp321 1110 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ( ph  /\  ps  /\  ch )  /\  ta )
 )  ->  ph )
 
Theoremsimp322 1111 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ( ph  /\  ps  /\  ch )  /\  ta )
 )  ->  ps )
 
Theoremsimp323 1112 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ( ph  /\  ps  /\  ch )  /\  ta )
 )  ->  ch )
 
Theoremsimp331 1113 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ta 
 /\  ( ph  /\  ps  /\ 
 ch ) ) ) 
 ->  ph )
 
Theoremsimp332 1114 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ta 
 /\  ( ph  /\  ps  /\ 
 ch ) ) ) 
 ->  ps )
 
Theoremsimp333 1115 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ta 
 /\  ( ph  /\  ps  /\ 
 ch ) ) ) 
 ->  ch )
 
Theorem3adantl1 1116 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ta 
 /\  ph  /\  ps )  /\  ch )  ->  th )
 
Theorem3adantl2 1117 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ta  /\  ps )  /\  ch )  ->  th )
 
Theorem3adantl3 1118 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ps  /\  ta )  /\  ch )  ->  th )
 
Theorem3adantr1 1119 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ta  /\  ps  /\  ch ) )  ->  th )
 
Theorem3adantr2 1120 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ta  /\  ch ) )  ->  th )
 
Theorem3adantr3 1121 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ch  /\  ta ) )  ->  th )
 
Theorem3ad2antl1 1122 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ps  /\  ta )  /\  ch )  ->  th )
 
Theorem3ad2antl2 1123 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ( ps 
 /\  ph  /\  ta )  /\  ch )  ->  th )
 
Theorem3ad2antl3 1124 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ( ps 
 /\  ta  /\  ph )  /\  ch )  ->  th )
 
Theorem3ad2antr1 1125 Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ch  /\  ps  /\  ta ) )  ->  th )
 
Theorem3ad2antr2 1126 Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ch  /\  ta ) )  ->  th )
 
Theorem3ad2antr3 1127 Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ta  /\  ch ) )  ->  th )
 
Theorem3anibar 1128 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 ) )
 
Theorem3mix1 1129 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ph  ->  ( ph  \/  ps  \/  ch ) )
 
Theorem3mix2 1130 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ph  ->  ( ps  \/  ph  \/  ch )
 )
 
Theorem3mix3 1131 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ph  ->  ( ps  \/  ch  \/  ph ) )
 
Theorem3mix1i 1132 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ph   =>    |-  ( ph  \/  ps  \/  ch )
 
Theorem3mix2i 1133 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ph   =>    |-  ( ps  \/  ph  \/  ch )
 
Theorem3mix3i 1134 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ph   =>    |-  ( ps  \/  ch  \/  ph )
 
Theorem3pm3.2i 1135 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
 |-  ph   &    |- 
 ps   &    |- 
 ch   =>    |-  ( ph  /\  ps  /\ 
 ch )
 
Theorempm3.2an3 1136 pm3.2 436 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  (
 ph  /\  ps  /\  ch ) ) ) )
 
Theorem3jca 1137 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  ( ps  /\  ch  /\  th ) )
 
Theorem3jcad 1138 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 ) ) )
 
Theoremmpbir3an 1139 Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.)
 |- 
 ps   &    |- 
 ch   &    |- 
 th   &    |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ph
 
Theoremmpbir3and 1140 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 )
 
Theoremsyl3anbrc 1141 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ta  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ta )
 
Theorem3anim123i 1142 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ta  ->  et )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  ->  ( ps  /\  th  /\  et ) )
 
Theorem3anim1i 1143 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  ch  /\  th )  ->  ( ps  /\ 
 ch  /\  th )
 )
 
Theorem3anim3i 1144 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  th  /\  ph )  ->  ( ch 
 /\  th  /\  ps )
 )
 
Theorem3anbi123i 1145 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ta  <->  et )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  <->  ( ps  /\  th 
 /\  et ) )
 
Theorem3orbi123i 1146 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ta  <->  et )   =>    |-  ( ( ph  \/  ch 
 \/  ta )  <->  ( ps  \/  th 
 \/  et ) )
 
Theorem3anbi1i 1147 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  /\  ch  /\ 
 th )  <->  ( ps  /\  ch 
 /\  th ) )
 
Theorem3anbi2i 1148 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  /\  ph 
 /\  th )  <->  ( ch  /\  ps 
 /\  th ) )
 
Theorem3anbi3i 1149 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  /\  th 
 /\  ph )  <->  ( ch  /\  th 
 /\  ps ) )
 
Theorem3imp 1150 Importation inference. (Contributed by NM, 8-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorem3impa 1151 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem3impb 1152 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem3impia 1153 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem3impib 1154 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorem3exp 1155 Exportation inference. (Contributed by NM, 30-May-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )
 
Theorem3expa 1156 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ph  /\  ps )  /\  ch )  ->  th )
 
Theorem3expb 1157 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  ->  th )
 
Theorem3expia 1158 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps )  ->  ( ch  ->  th )
 )
 
Theorem3expib 1159 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  ( ph  ->  ( ( ps 
 /\  ch )  ->  th )
 )
 
Theorem3com12 1160 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 )
 
Theorem3com13 1161 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ch  /\  ps  /\  ph )  ->  th )
 
Theorem3com23 1162 Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ch  /\  ps )  ->  th )
 
Theorem3coml 1163 Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ps  /\  ch  /\  ph )  ->  th )
 
Theorem3comr 1164 Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ch  /\  ph  /\  ps )  ->  th )
 
Theorem3adant3r1 1165 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ta 
 /\  ps  /\  ch )
 )  ->  th )
 
Theorem3adant3r2 1166 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ta  /\  ch )
 )  ->  th )
 
Theorem3adant3r3 1167 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch  /\  ta )
 )  ->  th )
 
Theorem3an1rs 1168 Swap conjuncts. (Contributed by NM, 16-Dec-2007.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ps  /\ 
 th )  /\  ch )  ->  ta )
 
Theorem3imp1 1169 Importation to left triple conjunction. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( (
 ph  /\  ps  /\  ch )  /\  th )  ->  ta )
 
Theorem3impd 1170 Importation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch  /\ 
 th )  ->  ta )
 )
 
Theorem3imp2 1171 Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\  ( ps  /\  ch  /\ 
 th ) )  ->  ta )
 
Theorem3exp1 1172 Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) ) )
 
Theorem3expd 1173 Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ph  ->  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorem3exp2 1174 Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) ) )
 
Theoremexp5o 1175 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  (
 ( th  /\  ta )  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp516 1176 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ( ( ph  /\  ( ps  /\  ch  /\ 
 th ) )  /\  ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp520 1177 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  ( th  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theorem3anassrs 1178 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )
 
Theorem3adant1l 1179 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ta  /\  ph )  /\  ps  /\  ch )  ->  th )
 
Theorem3adant1r 1180 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ph  /\  ta )  /\  ps  /\  ch )  ->  th )
 
Theorem3adant2l 1181 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ta 
 /\  ps )  /\  ch )  ->  th )
 
Theorem3adant2r 1182 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ta )  /\  ch )  ->  th )
 
Theorem3adant3l 1183 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps  /\  ( ta  /\  ch )
 )  ->  th )
 
Theorem3adant3r 1184 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps  /\  ( ch  /\  ta )
 )  ->  th )
 
Theoremsyl12anc 1185 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ( ps 
 /\  ( ch  /\  th ) )  ->  ta )   =>    |-  ( ph  ->  ta )
 
Theoremsyl21anc 1186 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ( ( ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ta )
 
Theoremsyl3anc 1187 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ph  ->  ta )
 
Theoremsyl22anc 1188 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl13anc 1189 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ps 
 /\  ( ch  /\  th 
 /\  ta ) )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl31anc 1190 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ta )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl112anc 1191 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ps 
 /\  ch  /\  ( th  /\ 
 ta ) )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl121anc 1192 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ps 
 /\  ( ch  /\  th )  /\  ta )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl211anc 1193 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ( ps  /\  ch )  /\  th  /\  ta )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl23anc 1194 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\ 
 et ) )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl32anc 1195 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et ) )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl122anc 1196 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ps 
 /\  ( ch  /\  th )  /\  ( ta 
 /\  et ) )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl212anc 1197 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ( ps  /\  ch )  /\  th  /\  ( ta 
 /\  et ) )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl221anc 1198 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta )  /\  et )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl113anc 1199 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ps 
 /\  ch  /\  ( th  /\ 
 ta  /\  et )
 )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl131anc 1200 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ps 
 /\  ( ch  /\  th 
 /\  ta )  /\  et )  ->  ze )   =>    |-  ( ph  ->  ze )
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