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Type | Label | Description |
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Statement | ||
Theorem | syl333anc 1201 | A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → 𝜏) & ⊢ (𝜑 → 𝜂) & ⊢ (𝜑 → 𝜁) & ⊢ (𝜑 → 𝜎) & ⊢ (𝜑 → 𝜌) & ⊢ (𝜑 → 𝜇) & ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌 ∧ 𝜇)) → 𝜆) ⇒ ⊢ (𝜑 → 𝜆) | ||
Theorem | syl3an1 1202 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
⊢ (𝜑 → 𝜓) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) | ||
Theorem | syl3an2 1203 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
⊢ (𝜑 → 𝜒) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏) | ||
Theorem | syl3an3 1204 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
⊢ (𝜑 → 𝜃) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏) | ||
Theorem | syl3an1b 1205 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) | ||
Theorem | syl3an2b 1206 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
⊢ (𝜑 ↔ 𝜒) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏) | ||
Theorem | syl3an3b 1207 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
⊢ (𝜑 ↔ 𝜃) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏) | ||
Theorem | syl3an1br 1208 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
⊢ (𝜓 ↔ 𝜑) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) | ||
Theorem | syl3an2br 1209 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
⊢ (𝜒 ↔ 𝜑) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏) | ||
Theorem | syl3an3br 1210 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
⊢ (𝜃 ↔ 𝜑) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏) | ||
Theorem | syl3an 1211 | 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 1212 | 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 1213 | 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 | syld3an3 1214 | A syllogism inference. (Contributed by NM, 20-May-2007.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏) | ||
Theorem | syld3an1 1215 | A syllogism inference. (Contributed by NM, 7-Jul-2008.) |
⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜑) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜏) | ||
Theorem | syld3an2 1216 | A syllogism inference. (Contributed by NM, 20-May-2007.) |
⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜓) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) | ||
Theorem | syl3anl1 1217 | A syllogism inference. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝜑 → 𝜓) & ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) ⇒ ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) | ||
Theorem | syl3anl2 1218 | A syllogism inference. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝜑 → 𝜒) & ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) ⇒ ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜃) ∧ 𝜏) → 𝜂) | ||
Theorem | syl3anl3 1219 | A syllogism inference. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝜑 → 𝜃) & ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) ⇒ ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜑) ∧ 𝜏) → 𝜂) | ||
Theorem | syl3anl 1220 | A triple syllogism inference. (Contributed by NM, 24-Dec-2006.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ (𝜏 → 𝜂) & ⊢ (((𝜓 ∧ 𝜃 ∧ 𝜂) ∧ 𝜁) → 𝜎) ⇒ ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜏) ∧ 𝜁) → 𝜎) | ||
Theorem | syl3anr1 1221 | A syllogism inference. (Contributed by NM, 31-Jul-2007.) |
⊢ (𝜑 → 𝜓) & ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂) ⇒ ⊢ ((𝜒 ∧ (𝜑 ∧ 𝜃 ∧ 𝜏)) → 𝜂) | ||
Theorem | syl3anr2 1222 | A syllogism inference. (Contributed by NM, 1-Aug-2007.) |
⊢ (𝜑 → 𝜃) & ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂) ⇒ ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑 ∧ 𝜏)) → 𝜂) | ||
Theorem | syl3anr3 1223 | A syllogism inference. (Contributed by NM, 23-Aug-2007.) |
⊢ (𝜑 → 𝜏) & ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂) ⇒ ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜑)) → 𝜂) | ||
Theorem | 3impdi 1224 | Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.) |
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | 3impdir 1225 | Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.) |
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃) | ||
Theorem | 3anidm12 1226 | Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) |
⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | 3anidm13 1227 | Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | 3anidm23 1228 | Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | syl2an3an 1229 | syl3an 1211 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜃 → 𝜏) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜃) → 𝜂) | ||
Theorem | syl2an23an 1230 | Deduction related to syl3an 1211 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜃 ∧ 𝜑) → 𝜏) & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜃 ∧ 𝜑) → 𝜂) | ||
Theorem | 3ori 1231 | Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.) |
⊢ (𝜑 ∨ 𝜓 ∨ 𝜒) ⇒ ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒) | ||
Theorem | 3jao 1232 | Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)) | ||
Theorem | 3jaob 1233 | Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.) |
⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓))) | ||
Theorem | 3jaoi 1234 | Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜓) & ⊢ (𝜃 → 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) | ||
Theorem | 3jaod 1235 | Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜃 → 𝜒)) & ⊢ (𝜑 → (𝜏 → 𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜏) → 𝜒)) | ||
Theorem | 3jaoian 1236 | Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜃 ∧ 𝜓) → 𝜒) & ⊢ ((𝜏 ∧ 𝜓) → 𝜒) ⇒ ⊢ (((𝜑 ∨ 𝜃 ∨ 𝜏) ∧ 𝜓) → 𝜒) | ||
Theorem | 3jaodan 1237 | Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜃) → 𝜒) & ⊢ ((𝜑 ∧ 𝜏) → 𝜒) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃 ∨ 𝜏)) → 𝜒) | ||
Theorem | mpjao3dan 1238 | Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜃) → 𝜒) & ⊢ ((𝜑 ∧ 𝜏) → 𝜒) & ⊢ (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜏)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | 3jaao 1239 | Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → (𝜏 → 𝜒)) & ⊢ (𝜂 → (𝜁 → 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → ((𝜓 ∨ 𝜏 ∨ 𝜁) → 𝜒)) | ||
Theorem | 3ianorr 1240 | Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.) |
⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
Theorem | syl3an9b 1241 | Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜃 → (𝜒 ↔ 𝜏)) & ⊢ (𝜂 → (𝜏 ↔ 𝜁)) ⇒ ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → (𝜓 ↔ 𝜁)) | ||
Theorem | 3orbi123d 1242 | Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜃 ↔ 𝜏)) & ⊢ (𝜑 → (𝜂 ↔ 𝜁)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ (𝜒 ∨ 𝜏 ∨ 𝜁))) | ||
Theorem | 3anbi123d 1243 | Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜃 ↔ 𝜏)) & ⊢ (𝜑 → (𝜂 ↔ 𝜁)) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ (𝜒 ∧ 𝜏 ∧ 𝜁))) | ||
Theorem | 3anbi12d 1244 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ (𝜒 ∧ 𝜏 ∧ 𝜂))) | ||
Theorem | 3anbi13d 1245 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜂 ∧ 𝜃) ↔ (𝜒 ∧ 𝜂 ∧ 𝜏))) | ||
Theorem | 3anbi23d 1246 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → ((𝜂 ∧ 𝜓 ∧ 𝜃) ↔ (𝜂 ∧ 𝜒 ∧ 𝜏))) | ||
Theorem | 3anbi1d 1247 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜏) ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) | ||
Theorem | 3anbi2d 1248 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ((𝜃 ∧ 𝜓 ∧ 𝜏) ↔ (𝜃 ∧ 𝜒 ∧ 𝜏))) | ||
Theorem | 3anbi3d 1249 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒))) | ||
Theorem | 3anim123d 1250 | Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜃 → 𝜏)) & ⊢ (𝜑 → (𝜂 → 𝜁)) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) → (𝜒 ∧ 𝜏 ∧ 𝜁))) | ||
Theorem | 3orim123d 1251 | Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜃 → 𝜏)) & ⊢ (𝜑 → (𝜂 → 𝜁)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) → (𝜒 ∨ 𝜏 ∨ 𝜁))) | ||
Theorem | an6 1252 | Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏) ∧ (𝜒 ∧ 𝜂))) | ||
Theorem | 3an6 1253 | Analog of an4 550 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜏) ∧ (𝜓 ∧ 𝜃 ∧ 𝜂))) | ||
Theorem | 3or6 1254 | Analog of or4 720 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) |
⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃) ∨ (𝜏 ∨ 𝜂)) ↔ ((𝜑 ∨ 𝜒 ∨ 𝜏) ∨ (𝜓 ∨ 𝜃 ∨ 𝜂))) | ||
Theorem | mp3an1 1255 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ 𝜑 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | mp3an2 1256 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ 𝜓 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | ||
Theorem | mp3an3 1257 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | mp3an12 1258 | An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜒 → 𝜃) | ||
Theorem | mp3an13 1259 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
⊢ 𝜑 & ⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜓 → 𝜃) | ||
Theorem | mp3an23 1260 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
⊢ 𝜓 & ⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | mp3an1i 1261 | An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) |
⊢ 𝜓 & ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)) ⇒ ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏)) | ||
Theorem | mp3anl1 1262 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ 𝜑 & ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | ||
Theorem | mp3anl2 1263 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ 𝜓 & ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏) | ||
Theorem | mp3anl3 1264 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ 𝜒 & ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏) | ||
Theorem | mp3anr1 1265 | An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) |
⊢ 𝜓 & ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) ⇒ ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜏) | ||
Theorem | mp3anr2 1266 | An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.) |
⊢ 𝜒 & ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) → 𝜏) | ||
Theorem | mp3anr3 1267 | An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) |
⊢ 𝜃 & ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏) | ||
Theorem | mp3an 1268 | An inference based on modus ponens. (Contributed by NM, 14-May-1999.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ 𝜃 | ||
Theorem | mpd3an3 1269 | An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | mpd3an23 1270 | An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | mp3and 1271 | A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)) ⇒ ⊢ (𝜑 → 𝜏) | ||
Theorem | mp3an12i 1272 | mp3an 1268 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ (𝜒 → 𝜃) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜒 → 𝜏) | ||
Theorem | mp3an2i 1273 | mp3an 1268 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ (𝜓 → 𝜃) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜓 → 𝜏) | ||
Theorem | mp3an3an 1274 | mp3an 1268 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ (𝜃 → 𝜏) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜓 ∧ 𝜃) → 𝜂) | ||
Theorem | mp3an2ani 1275 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ ((𝜓 ∧ 𝜃) → 𝜏) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜓 ∧ 𝜃) → 𝜂) | ||
Theorem | biimp3a 1276 | Infer implication from a logical equivalence. Similar to biimpa 290. (Contributed by NM, 4-Sep-2005.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | biimp3ar 1277 | Infer implication from a logical equivalence. Similar to biimpar 291. (Contributed by NM, 2-Jan-2009.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒) | ||
Theorem | 3anandis 1278 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) |
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)) → 𝜏) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) | ||
Theorem | 3anandirs 1279 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.) |
⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜃) ∧ (𝜒 ∧ 𝜃)) → 𝜏) ⇒ ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | ||
Theorem | ecased 1280 | Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | ecase23d 1281 | Variation of ecased 1280 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → ¬ 𝜃) & ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Even though it isn't ordinarily part of propositional calculus, the universal quantifier ∀ is introduced here so that the soundness of definition df-tru 1287 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in axiom ax-5 1376 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1292 may be adopted and this subsection moved down to the start of the subsection with wex 1421 below. However, the use of dftru2 1292 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | wal 1282 | Extend wff definition to include the universal quantifier ('for all'). ∀𝑥𝜑 is read "𝜑 (phi) is true for all 𝑥." Typically, in its final application 𝜑 would be replaced with a wff containing a (free) occurrence of the variable 𝑥, for example 𝑥 = 𝑦. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of 𝑥. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same. |
wff ∀𝑥𝜑 | ||
Even though it isn't ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1287 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in axiom ax-8 1435 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1292 may be adopted and this subsection moved down to just above weq 1432 below. However, the use of dftru2 1292 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | cv 1283 |
This syntax construction states that a variable 𝑥, which has been
declared to be a setvar variable by $f statement vx, is also a class
expression. This can be justified informally as follows. We know that
the class builder {𝑦 ∣ 𝑦 ∈ 𝑥} is a class by cab 2067.
Since (when
𝑦 is distinct from 𝑥) we
have 𝑥 =
{𝑦 ∣ 𝑦 ∈ 𝑥} by
cvjust 2076, we can argue that the syntax "class 𝑥 " can be viewed as
an abbreviation for "class {𝑦 ∣ 𝑦 ∈ 𝑥}". See the discussion
under the definition of class in [Jech] p.
4 showing that "Every set can
be considered to be a class."
While it is tempting and perhaps occasionally useful to view cv 1283 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1283 is intrinsically no different from any other class-building syntax such as cab 2067, cun 2971, or c0 3251. For a general discussion of the theory of classes and the role of cv 1283, see http://us.metamath.org/mpeuni/mmset.html#class. (The description above applies to set theory, not predicate calculus. The purpose of introducing class 𝑥 here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1432 of predicate calculus from the wceq 1284 of set theory, so that we don't overload the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.) |
class 𝑥 | ||
Syntax | wceq 1284 |
Extend wff definition to include class equality.
For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (The purpose of introducing wff 𝐴 = 𝐵 here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1432 of predicate calculus in terms of the wceq 1284 of set theory, so that we don't "overload" the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the = in 𝑥 = 𝑦 could be the = of either weq 1432 or wceq 1284, although mathematically it makes no difference. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2074 for more information on the set theory usage of wceq 1284.) |
wff 𝐴 = 𝐵 | ||
Syntax | wtru 1285 | ⊤ is a wff. |
wff ⊤ | ||
Theorem | trujust 1286 | Soundness justification theorem for df-tru 1287. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) |
⊢ ((∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) ↔ (∀𝑦 𝑦 = 𝑦 → ∀𝑦 𝑦 = 𝑦)) | ||
Definition | df-tru 1287 | Definition of the truth value "true", or "verum", denoted by ⊤. This is a tautology, as proved by tru 1288. In this definition, an instance of id 19 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 19 instance was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1288, and other proofs should depend on tru 1288 (directly or indirectly) instead of this definition, since there are many alternate ways to define ⊤. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) (New usage is discouraged.) |
⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) | ||
Theorem | tru 1288 | The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.) |
⊢ ⊤ | ||
Syntax | wfal 1289 | ⊥ is a wff. |
wff ⊥ | ||
Definition | df-fal 1290 | Definition of the truth value "false", or "falsum", denoted by ⊥. See also df-tru 1287. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (⊥ ↔ ¬ ⊤) | ||
Theorem | fal 1291 | The truth value ⊥ is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) |
⊢ ¬ ⊥ | ||
Theorem | dftru2 1292 | An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.) |
⊢ (⊤ ↔ (𝜑 → 𝜑)) | ||
Theorem | trud 1293 | Eliminate ⊤ as an antecedent. A proposition implied by ⊤ is true. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ (⊤ → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | tbtru 1294 | A proposition is equivalent to itself being equivalent to ⊤. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (𝜑 ↔ (𝜑 ↔ ⊤)) | ||
Theorem | nbfal 1295 | The negation of a proposition is equivalent to itself being equivalent to ⊥. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥)) | ||
Theorem | bitru 1296 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ 𝜑 ⇒ ⊢ (𝜑 ↔ ⊤) | ||
Theorem | bifal 1297 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ ¬ 𝜑 ⇒ ⊢ (𝜑 ↔ ⊥) | ||
Theorem | falim 1298 | The truth value ⊥ implies anything. Also called the principle of explosion, or "ex falso quodlibet". (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
⊢ (⊥ → 𝜑) | ||
Theorem | falimd 1299 | The truth value ⊥ implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((𝜑 ∧ ⊥) → 𝜓) | ||
Theorem | a1tru 1300 | Anything implies ⊤. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
⊢ (𝜑 → ⊤) |
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