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Theorem List for Intuitionistic Logic Explorer - 1101-1200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsimp22l 1101 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜂) → 𝜑)

Theoremsimp22r 1102 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜂) → 𝜓)

Theoremsimp23l 1103 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜂) → 𝜑)

Theoremsimp23r 1104 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜂) → 𝜓)

Theoremsimp31l 1105 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ ((𝜑𝜓) ∧ 𝜒𝜃)) → 𝜑)

Theoremsimp31r 1106 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ ((𝜑𝜓) ∧ 𝜒𝜃)) → 𝜓)

Theoremsimp32l 1107 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃)) → 𝜑)

Theoremsimp32r 1108 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃)) → 𝜓)

Theoremsimp33l 1109 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜒𝜃 ∧ (𝜑𝜓))) → 𝜑)

Theoremsimp33r 1110 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏𝜂 ∧ (𝜒𝜃 ∧ (𝜑𝜓))) → 𝜓)

Theoremsimp111 1111 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂𝜁) → 𝜑)

Theoremsimp112 1112 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂𝜁) → 𝜓)

Theoremsimp113 1113 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂𝜁) → 𝜒)

Theoremsimp121 1114 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂𝜁) → 𝜑)

Theoremsimp122 1115 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂𝜁) → 𝜓)

Theoremsimp123 1116 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂𝜁) → 𝜒)

Theoremsimp131 1117 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂𝜁) → 𝜑)

Theoremsimp132 1118 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂𝜁) → 𝜓)

Theoremsimp133 1119 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂𝜁) → 𝜒)

Theoremsimp211 1120 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜑)

Theoremsimp212 1121 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜓)

Theoremsimp213 1122 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜒)

Theoremsimp221 1123 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜑)

Theoremsimp222 1124 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜓)

Theoremsimp223 1125 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜒)

Theoremsimp231 1126 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜑)

Theoremsimp232 1127 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜓)

Theoremsimp233 1128 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜒)

Theoremsimp311 1129 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜑)

Theoremsimp312 1130 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜓)

Theoremsimp313 1131 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜒)

Theoremsimp321 1132 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜑)

Theoremsimp322 1133 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜓)

Theoremsimp323 1134 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜒)

Theoremsimp331 1135 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜑)

Theoremsimp332 1136 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜓)

Theoremsimp333 1137 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜒)

Theorem3adantl1 1138 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜏𝜑𝜓) ∧ 𝜒) → 𝜃)

Theorem3adantl2 1139 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜏𝜓) ∧ 𝜒) → 𝜃)

Theorem3adantl3 1140 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)

Theorem3adantr1 1141 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)

Theorem3adantr2 1142 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)

Theorem3adantr3 1143 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)

((𝜑𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)

((𝜑𝜒) → 𝜃)       (((𝜓𝜑𝜏) ∧ 𝜒) → 𝜃)

((𝜑𝜒) → 𝜃)       (((𝜓𝜏𝜑) ∧ 𝜒) → 𝜃)

Theorem3ad2antr1 1147 Deduction adding a conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜒𝜓𝜏)) → 𝜃)

Theorem3ad2antr2 1148 Deduction adding a conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)

Theorem3ad2antr3 1149 Deduction adding a conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)

Theorem3anibar 1150 Remove a hypothesis from the second member of a biconditional. (Contributed by FL, 22-Jul-2008.)
((𝜑𝜓𝜒) → (𝜃 ↔ (𝜒𝜏)))       ((𝜑𝜓𝜒) → (𝜃𝜏))

Theorem3mix1 1151 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜑𝜓𝜒))

Theorem3mix2 1152 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜓𝜑𝜒))

Theorem3mix3 1153 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜓𝜒𝜑))

Theorem3mix1i 1154 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜑𝜓𝜒)

Theorem3mix2i 1155 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜓𝜑𝜒)

Theorem3mix3i 1156 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜓𝜒𝜑)

Theorem3mix1d 1157 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒𝜃))

Theorem3mix2d 1158 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓𝜃))

Theorem3mix3d 1159 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜒𝜃𝜓))

Theorem3pm3.2i 1160 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
𝜑    &   𝜓    &   𝜒       (𝜑𝜓𝜒)

Theorempm3.2an3 1161 pm3.2 138 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
(𝜑 → (𝜓 → (𝜒 → (𝜑𝜓𝜒))))

Theorem3jca 1162 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → (𝜓𝜒𝜃))

Theorem3jcad 1163 Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓𝜏))       (𝜑 → (𝜓 → (𝜒𝜃𝜏)))

Theoremmpbir3an 1164 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 1165 Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.)
(𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃𝜏)))       (𝜑𝜓)

Theoremsyl3anbrc 1166 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜏 ↔ (𝜓𝜒𝜃))       (𝜑𝜏)

Theorem3anim123i 1167 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) → (𝜓𝜃𝜂))

Theorem3anim1i 1168 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
(𝜑𝜓)       ((𝜑𝜒𝜃) → (𝜓𝜒𝜃))

Theorem3anim2i 1169 Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
(𝜑𝜓)       ((𝜒𝜑𝜃) → (𝜒𝜓𝜃))

Theorem3anim3i 1170 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑𝜓)       ((𝜒𝜃𝜑) → (𝜒𝜃𝜓))

Theorem3anbi123i 1171 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))

Theorem3orbi123i 1172 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))

Theorem3anbi1i 1173 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜑𝜒𝜃) ↔ (𝜓𝜒𝜃))

Theorem3anbi2i 1174 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜑𝜃) ↔ (𝜒𝜓𝜃))

Theorem3anbi3i 1175 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜃𝜑) ↔ (𝜒𝜃𝜓))

Theorem3imp 1176 Importation inference. (Contributed by NM, 8-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorem3impa 1177 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theorem3imp31 1178 The importation inference 3imp 1176 with commutation of the first and third conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜒𝜓𝜑) → 𝜃)

Theorem3imp231 1179 Importation inference. (Contributed by Alan Sare, 17-Oct-2017.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜓𝜒𝜑) → 𝜃)

Theorem3imp21 1180 The importation inference 3imp 1176 with commutation of the first and second conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.) (Revised to shorten 3com12 1189 by Wolf Lammen, 23-Jun-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜓𝜑𝜒) → 𝜃)

Theorem3impb 1181 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theorem3impia 1182 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorem3impib 1183 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
(𝜑 → ((𝜓𝜒) → 𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorem3exp 1184 Exportation inference. (Contributed by NM, 30-May-1994.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theorem3expa 1185 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)

Theorem3expb 1186 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)

Theorem3expia 1187 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))

Theorem3expib 1188 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))

Theorem3com12 1189 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜑𝜒) → 𝜃)

Theorem3com13 1190 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜒𝜓𝜑) → 𝜃)

Theorem3com23 1191 Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)

Theorem3coml 1192 Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜒𝜑) → 𝜃)

Theorem3comr 1193 Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜒𝜑𝜓) → 𝜃)

Theorem3adant3r1 1194 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)

Theorem3adant3r2 1195 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)

Theorem3adant3r3 1196 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)

Theoremad4ant123 1197 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜃)

Theoremad4ant124 1198 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜓) ∧ 𝜏) ∧ 𝜒) → 𝜃)

Theoremad4ant134 1199 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Theoremad4ant234 1200 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜏𝜑) ∧ 𝜓) ∧ 𝜒) → 𝜃)

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