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Theorem List for Intuitionistic Logic Explorer - 1001-1100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimp1r2 1001 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏𝜂) → 𝜓)
 
Theoremsimp1r3 1002 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏𝜂) → 𝜒)
 
Theoremsimp2l1 1003 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜑)
 
Theoremsimp2l2 1004 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜓)
 
Theoremsimp2l3 1005 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜒)
 
Theoremsimp2r1 1006 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜑)
 
Theoremsimp2r2 1007 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜓)
 
Theoremsimp2r3 1008 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜒)
 
Theoremsimp3l1 1009 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜑)
 
Theoremsimp3l2 1010 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜓)
 
Theoremsimp3l3 1011 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜒)
 
Theoremsimp3r1 1012 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜑)
 
Theoremsimp3r2 1013 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜓)
 
Theoremsimp3r3 1014 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜒)
 
Theoremsimp11l 1015 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜏𝜂) → 𝜑)
 
Theoremsimp11r 1016 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜏𝜂) → 𝜓)
 
Theoremsimp12l 1017 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜏𝜂) → 𝜑)
 
Theoremsimp12r 1018 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜏𝜂) → 𝜓)
 
Theoremsimp13l 1019 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜏𝜂) → 𝜑)
 
Theoremsimp13r 1020 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜏𝜂) → 𝜓)
 
Theoremsimp21l 1021 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜂) → 𝜑)
 
Theoremsimp21r 1022 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜂) → 𝜓)
 
Theoremsimp22l 1023 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜂) → 𝜑)
 
Theoremsimp22r 1024 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜂) → 𝜓)
 
Theoremsimp23l 1025 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜂) → 𝜑)
 
Theoremsimp23r 1026 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜂) → 𝜓)
 
Theoremsimp31l 1027 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ ((𝜑𝜓) ∧ 𝜒𝜃)) → 𝜑)
 
Theoremsimp31r 1028 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ ((𝜑𝜓) ∧ 𝜒𝜃)) → 𝜓)
 
Theoremsimp32l 1029 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃)) → 𝜑)
 
Theoremsimp32r 1030 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃)) → 𝜓)
 
Theoremsimp33l 1031 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜒𝜃 ∧ (𝜑𝜓))) → 𝜑)
 
Theoremsimp33r 1032 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜒𝜃 ∧ (𝜑𝜓))) → 𝜓)
 
Theoremsimp111 1033 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂𝜁) → 𝜑)
 
Theoremsimp112 1034 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂𝜁) → 𝜓)
 
Theoremsimp113 1035 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂𝜁) → 𝜒)
 
Theoremsimp121 1036 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂𝜁) → 𝜑)
 
Theoremsimp122 1037 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂𝜁) → 𝜓)
 
Theoremsimp123 1038 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂𝜁) → 𝜒)
 
Theoremsimp131 1039 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂𝜁) → 𝜑)
 
Theoremsimp132 1040 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂𝜁) → 𝜓)
 
Theoremsimp133 1041 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂𝜁) → 𝜒)
 
Theoremsimp211 1042 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜑)
 
Theoremsimp212 1043 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜓)
 
Theoremsimp213 1044 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜒)
 
Theoremsimp221 1045 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜑)
 
Theoremsimp222 1046 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜓)
 
Theoremsimp223 1047 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜒)
 
Theoremsimp231 1048 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜑)
 
Theoremsimp232 1049 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜓)
 
Theoremsimp233 1050 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜒)
 
Theoremsimp311 1051 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜑)
 
Theoremsimp312 1052 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜓)
 
Theoremsimp313 1053 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜒)
 
Theoremsimp321 1054 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜑)
 
Theoremsimp322 1055 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜓)
 
Theoremsimp323 1056 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜒)
 
Theoremsimp331 1057 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜑)
 
Theoremsimp332 1058 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜓)
 
Theoremsimp333 1059 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜒)
 
Theorem3adantl1 1060 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜏𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adantl2 1061 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜏𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adantl3 1062 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3adantr1 1063 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)
 
Theorem3adantr2 1064 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3adantr3 1065 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3ad2antl1 1066 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antl2 1067 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜓𝜑𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antl3 1068 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜓𝜏𝜑) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antr1 1069 Deduction adding a conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜒𝜓𝜏)) → 𝜃)
 
Theorem3ad2antr2 1070 Deduction adding a conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3ad2antr3 1071 Deduction adding a conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3anibar 1072 Remove a hypothesis from the second member of a biimplication. (Contributed by FL, 22-Jul-2008.)
((𝜑𝜓𝜒) → (𝜃 ↔ (𝜒𝜏)))       ((𝜑𝜓𝜒) → (𝜃𝜏))
 
Theorem3mix1 1073 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜑𝜓𝜒))
 
Theorem3mix2 1074 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜓𝜑𝜒))
 
Theorem3mix3 1075 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜓𝜒𝜑))
 
Theorem3mix1i 1076 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜑𝜓𝜒)
 
Theorem3mix2i 1077 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜓𝜑𝜒)
 
Theorem3mix3i 1078 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜓𝜒𝜑)
 
Theorem3mix1d 1079 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒𝜃))
 
Theorem3mix2d 1080 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓𝜃))
 
Theorem3mix3d 1081 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜒𝜃𝜓))
 
Theorem3pm3.2i 1082 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
𝜑    &   𝜓    &   𝜒       (𝜑𝜓𝜒)
 
Theorempm3.2an3 1083 pm3.2 126 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
(𝜑 → (𝜓 → (𝜒 → (𝜑𝜓𝜒))))
 
Theorem3jca 1084 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → (𝜓𝜒𝜃))
 
Theorem3jcad 1085 Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓𝜏))       (𝜑 → (𝜓 → (𝜒𝜃𝜏)))
 
Theoremmpbir3an 1086 Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 9-Jan-2015.)
𝜓    &   𝜒    &   𝜃    &   (𝜑 ↔ (𝜓𝜒𝜃))       𝜑
 
Theoremmpbir3and 1087 Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.)
(𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃𝜏)))       (𝜑𝜓)
 
Theoremsyl3anbrc 1088 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜏 ↔ (𝜓𝜒𝜃))       (𝜑𝜏)
 
Theorem3anim123i 1089 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) → (𝜓𝜃𝜂))
 
Theorem3anim1i 1090 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
(𝜑𝜓)       ((𝜑𝜒𝜃) → (𝜓𝜒𝜃))
 
Theorem3anim2i 1091 Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
(𝜑𝜓)       ((𝜒𝜑𝜃) → (𝜒𝜓𝜃))
 
Theorem3anim3i 1092 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑𝜓)       ((𝜒𝜃𝜑) → (𝜒𝜃𝜓))
 
Theorem3anbi123i 1093 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
 
Theorem3orbi123i 1094 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
 
Theorem3anbi1i 1095 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜑𝜒𝜃) ↔ (𝜓𝜒𝜃))
 
Theorem3anbi2i 1096 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜑𝜃) ↔ (𝜒𝜓𝜃))
 
Theorem3anbi3i 1097 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜃𝜑) ↔ (𝜒𝜃𝜓))
 
Theorem3imp 1098 Importation inference. (Contributed by NM, 8-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impa 1099 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impb 1100 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
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