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Theorem List for Intuitionistic Logic Explorer - 1101-1200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3anim1i 1101 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  ch  /\  th )  ->  ( ps  /\ 
 ch  /\  th )
 )
 
Theorem3anim2i 1102 Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ph  /\  th )  ->  ( ch  /\  ps 
 /\  th ) )
 
Theorem3anim3i 1103 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  th  /\  ph )  ->  ( ch 
 /\  th  /\  ps )
 )
 
Theorem3anbi123i 1104 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ta  <->  et )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  <->  ( ps  /\  th 
 /\  et ) )
 
Theorem3orbi123i 1105 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ta  <->  et )   =>    |-  ( ( ph  \/  ch 
 \/  ta )  <->  ( ps  \/  th 
 \/  et ) )
 
Theorem3anbi1i 1106 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 1107 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 1108 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 1109 Importation inference. (Contributed by NM, 8-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorem3impa 1110 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem3impb 1111 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem3impia 1112 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem3impib 1113 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorem3exp 1114 Exportation inference. (Contributed by NM, 30-May-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )
 
Theorem3expa 1115 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ph  /\  ps )  /\  ch )  ->  th )
 
Theorem3expb 1116 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  ->  th )
 
Theorem3expia 1117 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps )  ->  ( ch  ->  th )
 )
 
Theorem3expib 1118 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  ( ph  ->  ( ( ps 
 /\  ch )  ->  th )
 )
 
Theorem3com12 1119 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 1120 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ch  /\  ps  /\  ph )  ->  th )
 
Theorem3com23 1121 Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ch  /\  ps )  ->  th )
 
Theorem3coml 1122 Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ps  /\  ch  /\  ph )  ->  th )
 
Theorem3comr 1123 Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ch  /\  ph  /\  ps )  ->  th )
 
Theorem3adant3r1 1124 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ta 
 /\  ps  /\  ch )
 )  ->  th )
 
Theorem3adant3r2 1125 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ta  /\  ch )
 )  ->  th )
 
Theorem3adant3r3 1126 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch  /\  ta )
 )  ->  th )
 
Theorem3an1rs 1127 Swap conjuncts. (Contributed by NM, 16-Dec-2007.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ps  /\ 
 th )  /\  ch )  ->  ta )
 
Theorem3imp1 1128 Importation to left triple conjunction. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( (
 ph  /\  ps  /\  ch )  /\  th )  ->  ta )
 
Theorem3impd 1129 Importation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch  /\ 
 th )  ->  ta )
 )
 
Theorem3imp2 1130 Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\  ( ps  /\  ch  /\ 
 th ) )  ->  ta )
 
Theorem3exp1 1131 Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) ) )
 
Theorem3expd 1132 Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ph  ->  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorem3exp2 1133 Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) ) )
 
Theoremexp5o 1134 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  (
 ( th  /\  ta )  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp516 1135 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ( ( ph  /\  ( ps  /\  ch  /\ 
 th ) )  /\  ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp520 1136 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  ( th  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theorem3anassrs 1137 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 1138 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ta  /\  ph )  /\  ps  /\  ch )  ->  th )
 
Theorem3adant1r 1139 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ph  /\  ta )  /\  ps  /\  ch )  ->  th )
 
Theorem3adant2l 1140 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ta 
 /\  ps )  /\  ch )  ->  th )
 
Theorem3adant2r 1141 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ta )  /\  ch )  ->  th )
 
Theorem3adant3l 1142 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps  /\  ( ta  /\  ch )
 )  ->  th )
 
Theorem3adant3r 1143 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps  /\  ( ch  /\  ta )
 )  ->  th )
 
Theoremsyl12anc 1144 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ( ps 
 /\  ( ch  /\  th ) )  ->  ta )   =>    |-  ( ph  ->  ta )
 
Theoremsyl21anc 1145 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ( ( ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ta )
 
Theoremsyl3anc 1146 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ph  ->  ta )
 
Theoremsyl22anc 1147 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 1148 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 1149 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 1150 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 1151 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 1152 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 1153 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 1154 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 1155 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 1156 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 1157 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 1158 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 1159 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 )
 
Theoremsyl311anc 1160 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 )
 
Theoremsyl33anc 1161 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et  /\  ze )
 )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl222anc 1162 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta )  /\  ( et  /\  ze ) )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl123anc 1163 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ps 
 /\  ( ch  /\  th )  /\  ( ta 
 /\  et  /\  ze )
 )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl132anc 1164 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ps 
 /\  ( ch  /\  th 
 /\  ta )  /\  ( et  /\  ze ) ) 
 ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl213anc 1165 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch )  /\  th  /\  ( ta 
 /\  et  /\  ze )
 )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl231anc 1166 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\ 
 et )  /\  ze )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl312anc 1167 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ta  /\  ( et  /\  ze )
 )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl321anc 1168 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et )  /\  ze )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl133anc 1169 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ps 
 /\  ( ch  /\  th 
 /\  ta )  /\  ( et  /\  ze  /\  si ) )  ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl313anc 1170 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ta  /\  ( et  /\  ze  /\  si ) )  ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl331anc 1171 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et  /\  ze )  /\  si )  ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl223anc 1172 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta )  /\  ( et  /\  ze 
 /\  si ) )  ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl232anc 1173 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\ 
 et )  /\  ( ze  /\  si ) ) 
 ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl322anc 1174 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et )  /\  ( ze  /\  si ) ) 
 ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl233anc 1175 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( 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 )
 
Theoremsyl323anc 1176 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( 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 )
 
Theoremsyl332anc 1177 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( 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 )
 
Theoremsyl333anc 1178 A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.)
 |-  ( 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 )
 
Theoremsyl3an1 1179 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )   =>    |-  (
 ( ph  /\  ch  /\  th )  ->  ta )
 
Theoremsyl3an2 1180 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )   =>    |-  (
 ( ps  /\  ph  /\  th )  ->  ta )
 
Theoremsyl3an3 1181 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  ->  th )   &    |-  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )   =>    |-  (
 ( ps  /\  ch  /\  ph )  ->  ta )
 
Theoremsyl3an1b 1182 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  <->  ps )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ph  /\  ch  /\ 
 th )  ->  ta )
 
Theoremsyl3an2b 1183 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  <->  ch )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ph 
 /\  th )  ->  ta )
 
Theoremsyl3an3b 1184 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  <->  th )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ch 
 /\  ph )  ->  ta )
 
Theoremsyl3an1br 1185 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ps  <->  ph )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ph  /\  ch  /\ 
 th )  ->  ta )
 
Theoremsyl3an2br 1186 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ch  <->  ph )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ph 
 /\  th )  ->  ta )
 
Theoremsyl3an3br 1187 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( th  <->  ph )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ch 
 /\  ph )  ->  ta )
 
Theoremsyl3an 1188 A triple syllogism inference. (Contributed by NM, 13-May-2004.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ta  ->  et )   &    |-  ( ( ps 
 /\  th  /\  et )  ->  ze )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  ->  ze )
 
Theoremsyl3anb 1189 A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ta  <->  et )   &    |-  ( ( ps 
 /\  th  /\  et )  ->  ze )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  ->  ze )
 
Theoremsyl3anbr 1190 A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)
 |-  ( ps  <->  ph )   &    |-  ( th  <->  ch )   &    |-  ( et  <->  ta )   &    |-  ( ( ps 
 /\  th  /\  et )  ->  ze )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  ->  ze )
 
Theoremsyld3an3 1191 A syllogism inference. (Contributed by NM, 20-May-2007.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  ta )
 
Theoremsyld3an1 1192 A syllogism inference. (Contributed by NM, 7-Jul-2008.)
 |-  ( ( ch  /\  ps 
 /\  th )  ->  ph )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  (
 ( ch  /\  ps  /\ 
 th )  ->  ta )
 
Theoremsyld3an2 1193 A syllogism inference. (Contributed by NM, 20-May-2007.)
 |-  ( ( ph  /\  ch  /\ 
 th )  ->  ps )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  (
 ( ph  /\  ch  /\  th )  ->  ta )
 
Theoremsyl3anl1 1194 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ( ps  /\  ch 
 /\  th )  /\  ta )  ->  et )   =>    |-  ( ( (
 ph  /\  ch  /\  th )  /\  ta )  ->  et )
 
Theoremsyl3anl2 1195 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ( ps  /\  ch 
 /\  th )  /\  ta )  ->  et )   =>    |-  ( ( ( ps  /\  ph  /\  th )  /\  ta )  ->  et )
 
Theoremsyl3anl3 1196 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  th )   &    |-  (
 ( ( ps  /\  ch 
 /\  th )  /\  ta )  ->  et )   =>    |-  ( ( ( ps  /\  ch  /\  ph )  /\  ta )  ->  et )
 
Theoremsyl3anl 1197 A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ta  ->  et )   &    |-  ( ( ( ps  /\  th  /\  et )  /\  ze )  ->  si )   =>    |-  ( ( ( ph  /\ 
 ch  /\  ta )  /\  ze )  ->  si )
 
Theoremsyl3anr1 1198 A syllogism inference. (Contributed by NM, 31-Jul-2007.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ph  /\  th  /\  ta ) )  ->  et )
 
Theoremsyl3anr2 1199 A syllogism inference. (Contributed by NM, 1-Aug-2007.)
 |-  ( ph  ->  th )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ps  /\  ph  /\  ta )
 )  ->  et )
 
Theoremsyl3anr3 1200 A syllogism inference. (Contributed by NM, 23-Aug-2007.)
 |-  ( ph  ->  ta )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ps  /\  th  /\  ph )
 )  ->  et )
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