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Theorem List for Metamath Proof Explorer - 1101-1200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3anidm 1101 Idempotent law for conjunction. (Contributed by Peter Mazsa, 17-Oct-2023.)
((𝜑𝜑𝜑) ↔ 𝜑)
 
Theorem3an4anass 1102 Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
(((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
 
Theorem3ioran 1103 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
(¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
 
Theorem3ianor 1104 Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Revised by Wolf Lammen, 8-Apr-2022.)
(¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
 
Theorem3anor 1105 Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Wolf Lammen, 8-Apr-2022.)
((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
 
Theorem3oran 1106 Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.)
((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
 
Theorem3impa 1107 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten 3imp 1108 by Wolf Lammen, 20-Jun-2022.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3imp 1108 Importation inference. (Contributed by NM, 8-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jun-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3imp31 1109 The importation inference 3imp 1108 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 1110 Importation inference. (Contributed by Alan Sare, 17-Oct-2017.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜓𝜒𝜑) → 𝜃)
 
Theorem3imp21 1111 The importation inference 3imp 1108 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 1120 by Wolf Lammen, 23-Jun-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜓𝜑𝜒) → 𝜃)
 
Theorem3impb 1112 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impib 1113 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
(𝜑 → ((𝜓𝜒) → 𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impia 1114 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3expa 1115 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten 3exp 1116 and pm3.2an3 1337 by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3exp 1116 Exportation inference. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theorem3expb 1117 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theorem3expia 1118 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))
 
Theorem3expib 1119 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theorem3com12 1120 Commutation in antecedent. Swap 1st and 2nd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜑𝜒) → 𝜃)
 
Theorem3com13 1121 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜒𝜓𝜑) → 𝜃)
 
Theorem3comr 1122 Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.) (Revised by Wolf Lammen, 9-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜒𝜑𝜓) → 𝜃)
 
Theorem3com23 1123 Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Wolf Lammen, 9-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)
 
Theorem3coml 1124 Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜒𝜑) → 𝜃)
 
Theorem3jca 1125 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → (𝜓𝜒𝜃))
 
Theorem3jcad 1126 Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓𝜏))       (𝜑 → (𝜓 → (𝜒𝜃𝜏)))
 
Theorem3adant1 1127 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
((𝜑𝜓) → 𝜒)       ((𝜃𝜑𝜓) → 𝜒)
 
Theorem3adant2 1128 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜑𝜃𝜓) → 𝜒)
 
Theorem3adant3 1129 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
((𝜑𝜓) → 𝜒)       ((𝜑𝜓𝜃) → 𝜒)
 
Theorem3ad2ant1 1130 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜑𝜓𝜃) → 𝜒)
 
Theorem3ad2ant2 1131 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜑𝜃) → 𝜒)
 
Theorem3ad2ant3 1132 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜃𝜑) → 𝜒)
 
Theoremsimp1 1133 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜑)
 
Theoremsimp2 1134 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜓)
 
Theoremsimp3 1135 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜒)
 
Theoremsimp1i 1136 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜑
 
Theoremsimp2i 1137 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜓
 
Theoremsimp3i 1138 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜒
 
Theoremsimp1d 1139 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜓)
 
Theoremsimp2d 1140 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜒)
 
Theoremsimp3d 1141 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜃)
 
Theoremsimp1bi 1142 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜓)
 
Theoremsimp2bi 1143 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜒)
 
Theoremsimp3bi 1144 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜃)
 
Theorem3simpa 1145 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
((𝜑𝜓𝜒) → (𝜑𝜓))
 
Theorem3simpb 1146 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
((𝜑𝜓𝜒) → (𝜑𝜒))
 
Theorem3simpc 1147 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
((𝜑𝜓𝜒) → (𝜓𝜒))
 
Theorem3anim123i 1148 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) → (𝜓𝜃𝜂))
 
Theorem3anim1i 1149 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
(𝜑𝜓)       ((𝜑𝜒𝜃) → (𝜓𝜒𝜃))
 
Theorem3anim2i 1150 Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
(𝜑𝜓)       ((𝜒𝜑𝜃) → (𝜒𝜓𝜃))
 
Theorem3anim3i 1151 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑𝜓)       ((𝜒𝜃𝜑) → (𝜒𝜃𝜓))
 
Theorem3anbi123i 1152 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
 
Theorem3orbi123i 1153 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
 
Theorem3anbi1i 1154 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜑𝜒𝜃) ↔ (𝜓𝜒𝜃))
 
Theorem3anbi2i 1155 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜑𝜃) ↔ (𝜒𝜓𝜃))
 
Theorem3anbi3i 1156 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜃𝜑) ↔ (𝜒𝜃𝜓))
 
Theoremsyl3an 1157 A triple syllogism inference. (Contributed by NM, 13-May-2004.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)
 
Theoremsyl3anb 1158 A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)
 
Theoremsyl3anbr 1159 A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)
(𝜓𝜑)    &   (𝜃𝜒)    &   (𝜂𝜏)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)
 
Theoremsyl3an1 1160 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜓)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)
 
Theoremsyl3an2 1161 A syllogism inference. (Contributed by NM, 22-Aug-1995.) (Proof shortened by Wolf Lammen, 26-Jun-2022.)
(𝜑𝜒)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)
 
Theoremsyl3an3 1162 A syllogism inference. (Contributed by NM, 22-Aug-1995.) (Proof shortened by Wolf Lammen, 26-Jun-2022.)
(𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)
 
Theorem3adantl1 1163 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜏𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adantl2 1164 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜏𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adantl3 1165 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3adantr1 1166 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)
 
Theorem3adantr2 1167 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3adantr3 1168 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theoremad4ant123 1169 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 1170 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 1171 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 1172 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)
 
Theorem3adant1l 1173 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜏𝜑) ∧ 𝜓𝜒) → 𝜃)
 
Theorem3adant1r 1174 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜑𝜏) ∧ 𝜓𝜒) → 𝜃)
 
Theorem3adant2l 1175 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜏𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adant2r 1176 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3adant3l 1177 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓 ∧ (𝜏𝜒)) → 𝜃)
 
Theorem3adant3r 1178 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓 ∧ (𝜒𝜏)) → 𝜃)
 
Theorem3adant3r1 1179 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)
 
Theorem3adant3r2 1180 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3adant3r3 1181 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3ad2antl1 1182 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antl2 1183 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜓𝜑𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antl3 1184 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜓𝜏𝜑) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antr1 1185 Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜒𝜓𝜏)) → 𝜃)
 
Theorem3ad2antr2 1186 Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3ad2antr3 1187 Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theoremsimpl1 1188 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜑)
 
Theoremsimpl2 1189 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimpl3 1190 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜒)
 
Theoremsimpr1 1191 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜓)
 
Theoremsimpr2 1192 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜒)
 
Theoremsimpr3 1193 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜃)
 
Theoremsimp1l 1194 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜑)
 
Theoremsimp1r 1195 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜓)
 
Theoremsimp2l 1196 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimp2r 1197 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜒)
 
Theoremsimp3l 1198 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜒)
 
Theoremsimp3r 1199 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜃)
 
Theoremsimp11 1200 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜑)
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