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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | anandi3 1101 | Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒))) | ||
Theorem | anandi3r 1102 | Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓))) | ||
Theorem | 3anidm 1103 | Idempotent law for conjunction. (Contributed by Peter Mazsa, 17-Oct-2023.) |
⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) ↔ 𝜑) | ||
Theorem | 3an4anass 1104 | Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃))) | ||
Theorem | 3ioran 1105 | Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.) |
⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒)) | ||
Theorem | 3ianor 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.) |
⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒)) | ||
Theorem | 3anor 1107 | Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Wolf Lammen, 8-Apr-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒)) | ||
Theorem | 3oran 1108 | Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒)) | ||
Theorem | 3impa 1109 | Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten 3imp 1110 by Wolf Lammen, 20-Jun-2022.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | 3imp 1110 | Importation inference. (Contributed by NM, 8-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jun-2022.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | 3imp31 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 → (𝜒 → 𝜃))) ⇒ ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜑) → 𝜃) | ||
Theorem | 3imp231 1112 | Importation inference. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) ⇒ ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜃) | ||
Theorem | 3imp21 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 → (𝜒 → 𝜃))) ⇒ ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃) | ||
Theorem | 3impb 1114 | Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | 3impib 1115 | Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) |
⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | 3impia 1116 | Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | 3expa 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.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | ||
Theorem | 3exp 1118 | Exportation inference. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) | ||
Theorem | 3expb 1119 | Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) | ||
Theorem | 3expia 1120 | Exportation from triple conjunction. (Contributed by NM, 19-May-2007.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) | ||
Theorem | 3expib 1121 | Exportation from triple conjunction. (Contributed by NM, 19-May-2007.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) | ||
Theorem | 3com12 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.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃) | ||
Theorem | 3com13 1123 | Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜑) → 𝜃) | ||
Theorem | 3comr 1124 | Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.) (Revised by Wolf Lammen, 9-Apr-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | 3com23 1125 | Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Wolf Lammen, 9-Apr-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃) | ||
Theorem | 3coml 1126 | Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜃) | ||
Theorem | 3jca 1127 | Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | 3jcad 1128 | Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (𝜓 → 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏))) | ||
Theorem | 3adant1 1129 | Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜃 ∧ 𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | 3adant2 1130 | Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜒) | ||
Theorem | 3adant3 1131 | Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒) | ||
Theorem | 3ad2ant1 1132 | Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.) |
⊢ (𝜑 → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒) | ||
Theorem | 3ad2ant2 1133 | Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.) |
⊢ (𝜑 → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜒) | ||
Theorem | 3ad2ant3 1134 | Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.) |
⊢ (𝜑 → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜑) → 𝜒) | ||
Theorem | simp1 1135 | Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜑) | ||
Theorem | simp2 1136 | Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓) | ||
Theorem | simp3 1137 | Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒) | ||
Theorem | simp1i 1138 | Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.) |
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒) ⇒ ⊢ 𝜑 | ||
Theorem | simp2i 1139 | Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.) |
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒) ⇒ ⊢ 𝜓 | ||
Theorem | simp3i 1140 | Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.) |
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒) ⇒ ⊢ 𝜒 | ||
Theorem | simp1d 1141 | Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.) |
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | simp2d 1142 | Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.) |
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | simp3d 1143 | Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.) |
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | simp1bi 1144 | Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | simp2bi 1145 | Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | simp3bi 1146 | Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | 3simpa 1147 | Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜓)) | ||
Theorem | 3simpb 1148 | Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 21-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒)) | ||
Theorem | 3simpc 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.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒)) | ||
Theorem | 3anim123i 1150 | Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ (𝜏 → 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂)) | ||
Theorem | 3anim1i 1151 | Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | 3anim2i 1152 | Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → (𝜒 ∧ 𝜓 ∧ 𝜃)) | ||
Theorem | 3anim3i 1153 | Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜑) → (𝜒 ∧ 𝜃 ∧ 𝜓)) | ||
Theorem | 3anbi123i 1154 | Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) & ⊢ (𝜏 ↔ 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) ↔ (𝜓 ∧ 𝜃 ∧ 𝜂)) | ||
Theorem | 3orbi123i 1155 | Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) & ⊢ (𝜏 ↔ 𝜂) ⇒ ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)) | ||
Theorem | 3anbi1i 1156 | Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | 3anbi2i 1157 | Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) ↔ (𝜒 ∧ 𝜓 ∧ 𝜃)) | ||
Theorem | 3anbi3i 1158 | Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜑) ↔ (𝜒 ∧ 𝜃 ∧ 𝜓)) | ||
Theorem | syl3an 1159 | A triple syllogism inference. (Contributed by NM, 13-May-2004.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ (𝜏 → 𝜂) & ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) | ||
Theorem | syl3anb 1160 | A triple syllogism inference. (Contributed by NM, 15-Oct-2005.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) & ⊢ (𝜏 ↔ 𝜂) & ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) | ||
Theorem | syl3anbr 1161 | A triple syllogism inference. (Contributed by NM, 29-Dec-2011.) |
⊢ (𝜓 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜒) & ⊢ (𝜂 ↔ 𝜏) & ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) | ||
Theorem | syl3an1 1162 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
⊢ (𝜑 → 𝜓) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) | ||
Theorem | syl3an2 1163 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) (Proof shortened by Wolf Lammen, 26-Jun-2022.) |
⊢ (𝜑 → 𝜒) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏) | ||
Theorem | syl3an3 1164 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) (Proof shortened by Wolf Lammen, 26-Jun-2022.) |
⊢ (𝜑 → 𝜃) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏) | ||
Theorem | 3adantl1 1165 | Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜏 ∧ 𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | ||
Theorem | 3adantl2 1166 | Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜑 ∧ 𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃) | ||
Theorem | 3adantl3 1167 | Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃) | ||
Theorem | 3adantr1 1168 | Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓 ∧ 𝜒)) → 𝜃) | ||
Theorem | 3adantr2 1169 | Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏 ∧ 𝜒)) → 𝜃) | ||
Theorem | 3adantr3 1170 | Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃) | ||
Theorem | ad4ant123 1171 | Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜃) | ||
Theorem | ad4ant124 1172 | Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜒) → 𝜃) | ||
Theorem | ad4ant134 1173 | Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃) | ||
Theorem | ad4ant234 1174 | Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((((𝜏 ∧ 𝜑) ∧ 𝜓) ∧ 𝜒) → 𝜃) | ||
Theorem | 3adant1l 1175 | Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | 3adant1r 1176 | Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜑 ∧ 𝜏) ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | 3adant2l 1177 | Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃) | ||
Theorem | 3adant2r 1178 | Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃) | ||
Theorem | 3adant3l 1179 | Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜏 ∧ 𝜒)) → 𝜃) | ||
Theorem | 3adant3r 1180 | Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜏)) → 𝜃) | ||
Theorem | 3adant3r1 1181 | Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓 ∧ 𝜒)) → 𝜃) | ||
Theorem | 3adant3r2 1182 | Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏 ∧ 𝜒)) → 𝜃) | ||
Theorem | 3adant3r3 1183 | Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃) | ||
Theorem | 3ad2antl1 1184 | Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.) |
⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃) | ||
Theorem | 3ad2antl2 1185 | Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.) |
⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜏) ∧ 𝜒) → 𝜃) | ||
Theorem | 3ad2antl3 1186 | Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.) |
⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((𝜓 ∧ 𝜏 ∧ 𝜑) ∧ 𝜒) → 𝜃) | ||
Theorem | 3ad2antr1 1187 | Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.) |
⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜓 ∧ 𝜏)) → 𝜃) | ||
Theorem | 3ad2antr2 1188 | Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.) |
⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃) | ||
Theorem | 3ad2antr3 1189 | Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.) |
⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏 ∧ 𝜒)) → 𝜃) | ||
Theorem | simpl1 1190 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜑) | ||
Theorem | simpl2 1191 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜓) | ||
Theorem | simpl3 1192 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜒) | ||
Theorem | simpr1 1193 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜓) | ||
Theorem | simpr2 1194 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜒) | ||
Theorem | simpr3 1195 | Simplification of conjunction. (Contributed by Jeff Hankins, 17-Nov-2009.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜃) | ||
Theorem | simp1l 1196 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) → 𝜑) | ||
Theorem | simp1r 1197 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) → 𝜓) | ||
Theorem | simp2l 1198 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜓) | ||
Theorem | simp2r 1199 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜒) | ||
Theorem | simp3l 1200 | Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜒) |
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