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Theorem List for Intuitionistic Logic Explorer - 1001-1100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem3adant3 1001 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜑𝜓𝜃) → 𝜒)

Theorem3ad2ant1 1002 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜑𝜓𝜃) → 𝜒)

Theorem3ad2ant2 1003 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜑𝜃) → 𝜒)

Theorem3ad2ant3 1004 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜃𝜑) → 𝜒)

Theoremsimp1l 1005 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜑)

Theoremsimp1r 1006 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜓)

Theoremsimp2l 1007 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜓)

Theoremsimp2r 1008 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜒)

Theoremsimp3l 1009 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜒)

Theoremsimp3r 1010 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜃)

Theoremsimp11 1011 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜑)

Theoremsimp12 1012 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜓)

Theoremsimp13 1013 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜒)

Theoremsimp21 1014 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜓)

Theoremsimp22 1015 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜒)

Theoremsimp23 1016 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜃)

Theoremsimp31 1017 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜒)

Theoremsimp32 1018 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜃)

Theoremsimp33 1019 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜏)

Theoremsimpll1 1020 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑)

Theoremsimpll2 1021 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)

Theoremsimpll3 1022 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜒)

Theoremsimplr1 1023 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏) → 𝜑)

Theoremsimplr2 1024 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏) → 𝜓)

Theoremsimplr3 1025 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏) → 𝜒)

Theoremsimprl1 1026 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜑)

Theoremsimprl2 1027 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜓)

Theoremsimprl3 1028 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜒)

Theoremsimprr1 1029 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜑)

Theoremsimprr2 1030 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜓)

Theoremsimprr3 1031 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜒)

Theoremsimpl1l 1032 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜏) → 𝜑)

Theoremsimpl1r 1033 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜏) → 𝜓)

Theoremsimpl2l 1034 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜏) → 𝜑)

Theoremsimpl2r 1035 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜏) → 𝜓)

Theoremsimpl3l 1036 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜏) → 𝜑)

Theoremsimpl3r 1037 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜏) → 𝜓)

Theoremsimpr1l 1038 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓) ∧ 𝜒𝜃)) → 𝜑)

Theoremsimpr1r 1039 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓) ∧ 𝜒𝜃)) → 𝜓)

Theoremsimpr2l 1040 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃)) → 𝜑)

Theoremsimpr2r 1041 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃)) → 𝜓)

Theoremsimpr3l 1042 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒𝜃 ∧ (𝜑𝜓))) → 𝜑)

Theoremsimpr3r 1043 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒𝜃 ∧ (𝜑𝜓))) → 𝜓)

Theoremsimp1ll 1044 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃𝜏) → 𝜑)

Theoremsimp1lr 1045 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃𝜏) → 𝜓)

Theoremsimp1rl 1046 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓)) ∧ 𝜃𝜏) → 𝜑)

Theoremsimp1rr 1047 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓)) ∧ 𝜃𝜏) → 𝜓)

Theoremsimp2ll 1048 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃 ∧ ((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜑)

Theoremsimp2lr 1049 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃 ∧ ((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜓)

Theoremsimp2rl 1050 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃 ∧ (𝜒 ∧ (𝜑𝜓)) ∧ 𝜏) → 𝜑)

Theoremsimp2rr 1051 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃 ∧ (𝜒 ∧ (𝜑𝜓)) ∧ 𝜏) → 𝜓)

Theoremsimp3ll 1052 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃𝜏 ∧ ((𝜑𝜓) ∧ 𝜒)) → 𝜑)

Theoremsimp3lr 1053 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃𝜏 ∧ ((𝜑𝜓) ∧ 𝜒)) → 𝜓)

Theoremsimp3rl 1054 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃𝜏 ∧ (𝜒 ∧ (𝜑𝜓))) → 𝜑)

Theoremsimp3rr 1055 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃𝜏 ∧ (𝜒 ∧ (𝜑𝜓))) → 𝜓)

Theoremsimpl11 1056 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂) → 𝜑)

Theoremsimpl12 1057 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂) → 𝜓)

Theoremsimpl13 1058 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂) → 𝜒)

Theoremsimpl21 1059 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜑)

Theoremsimpl22 1060 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜓)

Theoremsimpl23 1061 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜒)

Theoremsimpl31 1062 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜑)

Theoremsimpl32 1063 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜓)

Theoremsimpl33 1064 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜒)

Theoremsimpr11 1065 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜑)

Theoremsimpr12 1066 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜓)

Theoremsimpr13 1067 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜒)

Theoremsimpr21 1068 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜑)

Theoremsimpr22 1069 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜓)

Theoremsimpr23 1070 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜒)

Theoremsimpr31 1071 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜑)

Theoremsimpr32 1072 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜓)

Theoremsimpr33 1073 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜒)

Theoremsimp1l1 1074 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏𝜂) → 𝜑)

Theoremsimp1l2 1075 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏𝜂) → 𝜓)

Theoremsimp1l3 1076 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏𝜂) → 𝜒)

Theoremsimp1r1 1077 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏𝜂) → 𝜑)

Theoremsimp1r2 1078 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏𝜂) → 𝜓)

Theoremsimp1r3 1079 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏𝜂) → 𝜒)

Theoremsimp2l1 1080 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜑)

Theoremsimp2l2 1081 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜓)

Theoremsimp2l3 1082 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜒)

Theoremsimp2r1 1083 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜑)

Theoremsimp2r2 1084 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜓)

Theoremsimp2r3 1085 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜒)

Theoremsimp3l1 1086 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜑)

Theoremsimp3l2 1087 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜓)

Theoremsimp3l3 1088 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜒)

Theoremsimp3r1 1089 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜑)

Theoremsimp3r2 1090 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜓)

Theoremsimp3r3 1091 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜒)

Theoremsimp11l 1092 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜏𝜂) → 𝜑)

Theoremsimp11r 1093 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜏𝜂) → 𝜓)

Theoremsimp12l 1094 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜏𝜂) → 𝜑)

Theoremsimp12r 1095 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜏𝜂) → 𝜓)

Theoremsimp13l 1096 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜏𝜂) → 𝜑)

Theoremsimp13r 1097 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜏𝜂) → 𝜓)

Theoremsimp21l 1098 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜂) → 𝜑)

Theoremsimp21r 1099 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜂) → 𝜓)

Theoremsimp22l 1100 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜂) → 𝜑)

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