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Theorem List for Intuitionistic Logic Explorer - 1001-1100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimpr13 1001 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th  /\ 
 ta ) )  ->  ch )
 
Theoremsimpr21 1002 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta ) )  ->  ph )
 
Theoremsimpr22 1003 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta ) )  ->  ps )
 
Theoremsimpr23 1004 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta ) )  ->  ch )
 
Theoremsimpr31 1005 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) ) )  ->  ph )
 
Theoremsimpr32 1006 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) ) )  ->  ps )
 
Theoremsimpr33 1007 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) ) )  ->  ch )
 
Theoremsimp1l1 1008 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th )  /\  ta 
 /\  et )  ->  ph )
 
Theoremsimp1l2 1009 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th )  /\  ta 
 /\  et )  ->  ps )
 
Theoremsimp1l3 1010 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th )  /\  ta 
 /\  et )  ->  ch )
 
Theoremsimp1r1 1011 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch ) )  /\  ta 
 /\  et )  ->  ph )
 
Theoremsimp1r2 1012 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch ) )  /\  ta 
 /\  et )  ->  ps )
 
Theoremsimp1r3 1013 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch ) )  /\  ta 
 /\  et )  ->  ch )
 
Theoremsimp2l1 1014 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th )  /\  et )  ->  ph )
 
Theoremsimp2l2 1015 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th )  /\  et )  ->  ps )
 
Theoremsimp2l3 1016 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th )  /\  et )  ->  ch )
 
Theoremsimp2r1 1017 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et )  -> 
 ph )
 
Theoremsimp2r2 1018 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et )  ->  ps )
 
Theoremsimp2r3 1019 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et )  ->  ch )
 
Theoremsimp3l1 1020 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th ) )  ->  ph )
 
Theoremsimp3l2 1021 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th ) )  ->  ps )
 
Theoremsimp3l3 1022 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th ) )  ->  ch )
 
Theoremsimp3r1 1023 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( th  /\  ( ph  /\  ps  /\  ch ) ) )  ->  ph )
 
Theoremsimp3r2 1024 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( th  /\  ( ph  /\  ps  /\  ch ) ) )  ->  ps )
 
Theoremsimp3r3 1025 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( th  /\  ( ph  /\  ps  /\  ch ) ) )  ->  ch )
 
Theoremsimp11l 1026 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch  /\  th )  /\  ta  /\  et )  -> 
 ph )
 
Theoremsimp11r 1027 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch  /\  th )  /\  ta  /\  et )  ->  ps )
 
Theoremsimp12l 1028 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  ( ph  /\  ps )  /\  th )  /\  ta 
 /\  et )  ->  ph )
 
Theoremsimp12r 1029 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  ( ph  /\  ps )  /\  th )  /\  ta 
 /\  et )  ->  ps )
 
Theoremsimp13l 1030 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  th  /\  ( ph  /\ 
 ps ) )  /\  ta 
 /\  et )  ->  ph )
 
Theoremsimp13r 1031 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  th  /\  ( ph  /\ 
 ps ) )  /\  ta 
 /\  et )  ->  ps )
 
Theoremsimp21l 1032 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps )  /\  ch  /\  th )  /\  et )  ->  ph )
 
Theoremsimp21r 1033 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps )  /\  ch  /\  th )  /\  et )  ->  ps )
 
Theoremsimp22l 1034 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ch  /\  ( ph  /\ 
 ps )  /\  th )  /\  et )  ->  ph )
 
Theoremsimp22r 1035 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ch  /\  ( ph  /\ 
 ps )  /\  th )  /\  et )  ->  ps )
 
Theoremsimp23l 1036 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ch  /\  th  /\  ( ph  /\  ps )
 )  /\  et )  -> 
 ph )
 
Theoremsimp23r 1037 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ch  /\  th  /\  ( ph  /\  ps )
 )  /\  et )  ->  ps )
 
Theoremsimp31l 1038 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th ) )  ->  ph )
 
Theoremsimp31r 1039 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th ) )  ->  ps )
 
Theoremsimp32l 1040 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ch  /\  ( ph  /\  ps )  /\  th ) )  ->  ph )
 
Theoremsimp32r 1041 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ch  /\  ( ph  /\  ps )  /\  th ) )  ->  ps )
 
Theoremsimp33l 1042 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ch  /\  th 
 /\  ( ph  /\  ps ) ) )  ->  ph )
 
Theoremsimp33r 1043 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ch  /\  th 
 /\  ( ph  /\  ps ) ) )  ->  ps )
 
Theoremsimp111 1044 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th  /\  ta )  /\  et  /\  ze )  ->  ph )
 
Theoremsimp112 1045 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th  /\  ta )  /\  et  /\  ze )  ->  ps )
 
Theoremsimp113 1046 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th  /\  ta )  /\  et  /\  ze )  ->  ch )
 
Theoremsimp121 1047 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch )  /\  ta )  /\  et  /\  ze )  ->  ph )
 
Theoremsimp122 1048 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch )  /\  ta )  /\  et  /\  ze )  ->  ps )
 
Theoremsimp123 1049 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch )  /\  ta )  /\  et  /\  ze )  ->  ch )
 
Theoremsimp131 1050 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\ 
 ta  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et  /\  ze )  ->  ph )
 
Theoremsimp132 1051 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\ 
 ta  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et  /\  ze )  ->  ps )
 
Theoremsimp133 1052 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\ 
 ta  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et  /\  ze )  ->  ch )
 
Theoremsimp211 1053 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th  /\ 
 ta )  /\  ze )  ->  ph )
 
Theoremsimp212 1054 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th  /\ 
 ta )  /\  ze )  ->  ps )
 
Theoremsimp213 1055 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th  /\ 
 ta )  /\  ze )  ->  ch )
 
Theoremsimp221 1056 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta )  /\  ze )  ->  ph )
 
Theoremsimp222 1057 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta )  /\  ze )  ->  ps )
 
Theoremsimp223 1058 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta )  /\  ze )  ->  ch )
 
Theoremsimp231 1059 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) )  /\  ze )  ->  ph )
 
Theoremsimp232 1060 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) )  /\  ze )  ->  ps )
 
Theoremsimp233 1061 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) )  /\  ze )  ->  ch )
 
Theoremsimp311 1062 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )
 )  ->  ph )
 
Theoremsimp312 1063 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )
 )  ->  ps )
 
Theoremsimp313 1064 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )
 )  ->  ch )
 
Theoremsimp321 1065 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ( ph  /\  ps  /\  ch )  /\  ta )
 )  ->  ph )
 
Theoremsimp322 1066 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ( ph  /\  ps  /\  ch )  /\  ta )
 )  ->  ps )
 
Theoremsimp323 1067 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ( ph  /\  ps  /\  ch )  /\  ta )
 )  ->  ch )
 
Theoremsimp331 1068 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ta 
 /\  ( ph  /\  ps  /\ 
 ch ) ) ) 
 ->  ph )
 
Theoremsimp332 1069 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ta 
 /\  ( ph  /\  ps  /\ 
 ch ) ) ) 
 ->  ps )
 
Theoremsimp333 1070 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ta 
 /\  ( ph  /\  ps  /\ 
 ch ) ) ) 
 ->  ch )
 
Theorem3adantl1 1071 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ta 
 /\  ph  /\  ps )  /\  ch )  ->  th )
 
Theorem3adantl2 1072 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ta  /\  ps )  /\  ch )  ->  th )
 
Theorem3adantl3 1073 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ps  /\  ta )  /\  ch )  ->  th )
 
Theorem3adantr1 1074 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ta  /\  ps  /\  ch ) )  ->  th )
 
Theorem3adantr2 1075 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ta  /\  ch ) )  ->  th )
 
Theorem3adantr3 1076 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ch  /\  ta ) )  ->  th )
 
Theorem3ad2antl1 1077 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ps  /\  ta )  /\  ch )  ->  th )
 
Theorem3ad2antl2 1078 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ( ps 
 /\  ph  /\  ta )  /\  ch )  ->  th )
 
Theorem3ad2antl3 1079 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ( ps 
 /\  ta  /\  ph )  /\  ch )  ->  th )
 
Theorem3ad2antr1 1080 Deduction adding a conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ch  /\  ps  /\  ta ) )  ->  th )
 
Theorem3ad2antr2 1081 Deduction adding a conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ch  /\  ta ) )  ->  th )
 
Theorem3ad2antr3 1082 Deduction adding a conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ta  /\  ch ) )  ->  th )
 
Theorem3anibar 1083 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 1084 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ph  ->  ( ph  \/  ps  \/  ch ) )
 
Theorem3mix2 1085 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ph  ->  ( ps  \/  ph  \/  ch )
 )
 
Theorem3mix3 1086 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ph  ->  ( ps  \/  ch  \/  ph ) )
 
Theorem3mix1i 1087 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ph   =>    |-  ( ph  \/  ps  \/  ch )
 
Theorem3mix2i 1088 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ph   =>    |-  ( ps  \/  ph  \/  ch )
 
Theorem3mix3i 1089 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ph   =>    |-  ( ps  \/  ch  \/  ph )
 
Theorem3mix1d 1090 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch 
 \/  th ) )
 
Theorem3mix2d 1091 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps 
 \/  th ) )
 
Theorem3mix3d 1092 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  th 
 \/  ps ) )
 
Theorem3pm3.2i 1093 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
 |-  ph   &    |- 
 ps   &    |- 
 ch   =>    |-  ( ph  /\  ps  /\ 
 ch )
 
Theorempm3.2an3 1094 pm3.2 130 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  (
 ph  /\  ps  /\  ch ) ) ) )
 
Theorem3jca 1095 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  ( ps  /\  ch  /\  th ) )
 
Theorem3jcad 1096 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 1097 Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 9-Jan-2015.)
 |- 
 ps   &    |- 
 ch   &    |- 
 th   &    |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ph
 
Theoremmpbir3and 1098 Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  ( ps  <->  ( ch  /\  th 
 /\  ta ) ) )   =>    |-  ( ph  ->  ps )
 
Theoremsyl3anbrc 1099 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ta  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ta )
 
Theorem3anim123i 1100 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ta  ->  et )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  ->  ( ps  /\  th  /\  et ) )
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