HomeHome Metamath Proof Explorer
Theorem List (p. 12 of 465)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-29289)
  Hilbert Space Explorer  Hilbert Space Explorer
(29290-30812)
  Users' Mathboxes  Users' Mathboxes
(30813-46500)
 

Theorem List for Metamath Proof Explorer - 1101-1200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremanandi3 1101 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 
Theoremanandi3r 1102 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
 
Theorem3anidm 1103 Idempotent law for conjunction. (Contributed by Peter Mazsa, 17-Oct-2023.)
((𝜑𝜑𝜑) ↔ 𝜑)
 
Theorem3an4anass 1104 Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
(((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
 
Theorem3ioran 1105 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
(¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
 
Theorem3ianor 1106 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 1107 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 1108 Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.)
((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
 
Theorem3impa 1109 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten 3imp 1110 by Wolf Lammen, 20-Jun-2022.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3imp 1110 Importation inference. (Contributed by NM, 8-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jun-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3imp31 1111 The importation inference 3imp 1110 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 1112 Importation inference. (Contributed by Alan Sare, 17-Oct-2017.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜓𝜒𝜑) → 𝜃)
 
Theorem3imp21 1113 The importation inference 3imp 1110 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 1122 by Wolf Lammen, 23-Jun-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜓𝜑𝜒) → 𝜃)
 
Theorem3impb 1114 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impib 1115 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
(𝜑 → ((𝜓𝜒) → 𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impia 1116 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3expa 1117 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten 3exp 1118 and pm3.2an3 1339 by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3exp 1118 Exportation inference. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theorem3expb 1119 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theorem3expia 1120 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))
 
Theorem3expib 1121 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theorem3com12 1122 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 1123 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 1124 Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.) (Revised by Wolf Lammen, 9-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜒𝜑𝜓) → 𝜃)
 
Theorem3com23 1125 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 1126 Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜒𝜑) → 𝜃)
 
Theorem3jca 1127 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → (𝜓𝜒𝜃))
 
Theorem3jcad 1128 Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓𝜏))       (𝜑 → (𝜓 → (𝜒𝜃𝜏)))
 
Theorem3adant1 1129 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 1130 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜑𝜃𝜓) → 𝜒)
 
Theorem3adant3 1131 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 1132 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜑𝜓𝜃) → 𝜒)
 
Theorem3ad2ant2 1133 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜑𝜃) → 𝜒)
 
Theorem3ad2ant3 1134 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜃𝜑) → 𝜒)
 
Theoremsimp1 1135 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜑)
 
Theoremsimp2 1136 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜓)
 
Theoremsimp3 1137 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
((𝜑𝜓𝜒) → 𝜒)
 
Theoremsimp1i 1138 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜑
 
Theoremsimp2i 1139 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜓
 
Theoremsimp3i 1140 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜒
 
Theoremsimp1d 1141 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜓)
 
Theoremsimp2d 1142 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜒)
 
Theoremsimp3d 1143 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜃)
 
Theoremsimp1bi 1144 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜓)
 
Theoremsimp2bi 1145 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜒)
 
Theoremsimp3bi 1146 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜃)
 
Theorem3simpa 1147 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
((𝜑𝜓𝜒) → (𝜑𝜓))
 
Theorem3simpb 1148 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
((𝜑𝜓𝜒) → (𝜑𝜒))
 
Theorem3simpc 1149 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 1150 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) → (𝜓𝜃𝜂))
 
Theorem3anim1i 1151 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
(𝜑𝜓)       ((𝜑𝜒𝜃) → (𝜓𝜒𝜃))
 
Theorem3anim2i 1152 Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
(𝜑𝜓)       ((𝜒𝜑𝜃) → (𝜒𝜓𝜃))
 
Theorem3anim3i 1153 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑𝜓)       ((𝜒𝜃𝜑) → (𝜒𝜃𝜓))
 
Theorem3anbi123i 1154 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
 
Theorem3orbi123i 1155 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
 
Theorem3anbi1i 1156 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜑𝜒𝜃) ↔ (𝜓𝜒𝜃))
 
Theorem3anbi2i 1157 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜑𝜃) ↔ (𝜒𝜓𝜃))
 
Theorem3anbi3i 1158 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜃𝜑) ↔ (𝜒𝜃𝜓))
 
Theoremsyl3an 1159 A triple syllogism inference. (Contributed by NM, 13-May-2004.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)
 
Theoremsyl3anb 1160 A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)
 
Theoremsyl3anbr 1161 A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)
(𝜓𝜑)    &   (𝜃𝜒)    &   (𝜂𝜏)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)
 
Theoremsyl3an1 1162 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜓)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)
 
Theoremsyl3an2 1163 A syllogism inference. (Contributed by NM, 22-Aug-1995.) (Proof shortened by Wolf Lammen, 26-Jun-2022.)
(𝜑𝜒)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)
 
Theoremsyl3an3 1164 A syllogism inference. (Contributed by NM, 22-Aug-1995.) (Proof shortened by Wolf Lammen, 26-Jun-2022.)
(𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)
 
Theorem3adantl1 1165 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜏𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adantl2 1166 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜏𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adantl3 1167 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3adantr1 1168 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)
 
Theorem3adantr2 1169 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3adantr3 1170 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theoremad4ant123 1171 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 1172 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 1173 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 1174 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 1175 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 1176 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 1177 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 1178 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 1179 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 1180 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 1181 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)
 
Theorem3adant3r2 1182 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3adant3r3 1183 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3ad2antl1 1184 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antl2 1185 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜓𝜑𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antl3 1186 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜓𝜏𝜑) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antr1 1187 Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜒𝜓𝜏)) → 𝜃)
 
Theorem3ad2antr2 1188 Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3ad2antr3 1189 Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theoremsimpl1 1190 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜑)
 
Theoremsimpl2 1191 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimpl3 1192 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜒)
 
Theoremsimpr1 1193 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜓)
 
Theoremsimpr2 1194 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜒)
 
Theoremsimpr3 1195 Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜃)
 
Theoremsimp1l 1196 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜑)
 
Theoremsimp1r 1197 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜓)
 
Theoremsimp2l 1198 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimp2r 1199 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜒)
 
Theoremsimp3l 1200 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜒)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500
  Copyright terms: Public domain < Previous  Next >