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Theorem List for Intuitionistic Logic Explorer - 1201-1300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsyl233anc 1201 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)       (𝜑𝜇)

Theoremsyl323anc 1202 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)       (𝜑𝜇)

Theoremsyl332anc 1203 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌)) → 𝜇)       (𝜑𝜇)

Theoremsyl333anc 1204 A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (𝜑𝜇)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌𝜇)) → 𝜆)       (𝜑𝜆)

Theoremsyl3an1 1205 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜓)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)

Theoremsyl3an2 1206 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜒)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)

Theoremsyl3an3 1207 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)

Theoremsyl3an1b 1208 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜓)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)

Theoremsyl3an2b 1209 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜒)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)

Theoremsyl3an3b 1210 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)

Theoremsyl3an1br 1211 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜓𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)

Theoremsyl3an2br 1212 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜒𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)

Theoremsyl3an3br 1213 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜃𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)

Theoremsyl3an 1214 A triple syllogism inference. (Contributed by NM, 13-May-2004.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)

Theoremsyl3anb 1215 A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)

Theoremsyl3anbr 1216 A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)
(𝜓𝜑)    &   (𝜃𝜒)    &   (𝜂𝜏)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)

Theoremsyld3an3 1217 A syllogism inference. (Contributed by NM, 20-May-2007.)
((𝜑𝜓𝜒) → 𝜃)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜑𝜓𝜒) → 𝜏)

Theoremsyld3an1 1218 A syllogism inference. (Contributed by NM, 7-Jul-2008.)
((𝜒𝜓𝜃) → 𝜑)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜒𝜓𝜃) → 𝜏)

Theoremsyld3an2 1219 A syllogism inference. (Contributed by NM, 20-May-2007.)
((𝜑𝜒𝜃) → 𝜓)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)

Theoremsyl3anl1 1220 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(𝜑𝜓)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (((𝜑𝜒𝜃) ∧ 𝜏) → 𝜂)

Theoremsyl3anl2 1221 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(𝜑𝜒)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (((𝜓𝜑𝜃) ∧ 𝜏) → 𝜂)

Theoremsyl3anl3 1222 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(𝜑𝜃)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (((𝜓𝜒𝜑) ∧ 𝜏) → 𝜂)

Theoremsyl3anl 1223 A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   (((𝜓𝜃𝜂) ∧ 𝜁) → 𝜎)       (((𝜑𝜒𝜏) ∧ 𝜁) → 𝜎)

Theoremsyl3anr1 1224 A syllogism inference. (Contributed by NM, 31-Jul-2007.)
(𝜑𝜓)    &   ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)       ((𝜒 ∧ (𝜑𝜃𝜏)) → 𝜂)

Theoremsyl3anr2 1225 A syllogism inference. (Contributed by NM, 1-Aug-2007.)
(𝜑𝜃)    &   ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)       ((𝜒 ∧ (𝜓𝜑𝜏)) → 𝜂)

Theoremsyl3anr3 1226 A syllogism inference. (Contributed by NM, 23-Aug-2007.)
(𝜑𝜏)    &   ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)       ((𝜒 ∧ (𝜓𝜃𝜑)) → 𝜂)

Theorem3impdi 1227 Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theorem3impdir 1228 Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜓)) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)

Theorem3anidm12 1229 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
((𝜑𝜑𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theorem3anidm13 1230 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
((𝜑𝜓𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theorem3anidm23 1231 Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)
((𝜑𝜓𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremsyl2an3an 1232 syl3an 1214 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜃𝜏)    &   ((𝜓𝜒𝜏) → 𝜂)       ((𝜑𝜃) → 𝜂)

Theoremsyl2an23an 1233 Deduction related to syl3an 1214 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜃𝜑) → 𝜏)    &   ((𝜓𝜒𝜏) → 𝜂)       ((𝜃𝜑) → 𝜂)

Theorem3ori 1234 Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
(𝜑𝜓𝜒)       ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)

Theorem3jao 1235 Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.)
(((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)) → ((𝜑𝜒𝜃) → 𝜓))

Theorem3jaob 1236 Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.)
(((𝜑𝜒𝜃) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)))

Theorem3jaoi 1237 Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜃𝜓)       ((𝜑𝜒𝜃) → 𝜓)

Theorem3jaod 1238 Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜏𝜒))       (𝜑 → ((𝜓𝜃𝜏) → 𝜒))

Theorem3jaoian 1239 Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜃𝜓) → 𝜒)    &   ((𝜏𝜓) → 𝜒)       (((𝜑𝜃𝜏) ∧ 𝜓) → 𝜒)

Theorem3jaodan 1240 Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)    &   ((𝜑𝜏) → 𝜒)       ((𝜑 ∧ (𝜓𝜃𝜏)) → 𝜒)

Theoremmpjao3dan 1241 Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)    &   ((𝜑𝜏) → 𝜒)    &   (𝜑 → (𝜓𝜃𝜏))       (𝜑𝜒)

Theorem3jaao 1242 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 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))    &   (𝜂 → (𝜁𝜒))       ((𝜑𝜃𝜂) → ((𝜓𝜏𝜁) → 𝜒))

Theorem3ianorr 1243 Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.)
((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) → ¬ (𝜑𝜓𝜒))

Theoremsyl3an9b 1244 Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))    &   (𝜂 → (𝜏𝜁))       ((𝜑𝜃𝜂) → (𝜓𝜁))

Theorem3orbi123d 1245 Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜁)))

Theorem3anbi123d 1246 Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜁)))

Theorem3anbi12d 1247 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜂)))

Theorem3anbi13d 1248 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜂𝜃) ↔ (𝜒𝜂𝜏)))

Theorem3anbi23d 1249 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜂𝜓𝜃) ↔ (𝜂𝜒𝜏)))

Theorem3anbi1d 1250 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃𝜏) ↔ (𝜒𝜃𝜏)))

Theorem3anbi2d 1251 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓𝜏) ↔ (𝜃𝜒𝜏)))

Theorem3anbi3d 1252 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜏𝜓) ↔ (𝜃𝜏𝜒)))

Theorem3anim123d 1253 Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))

Theorem3orim123d 1254 Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))

Theoreman6 1255 Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏𝜂)) ↔ ((𝜑𝜃) ∧ (𝜓𝜏) ∧ (𝜒𝜂)))

Theorem3an6 1256 Analog of an4 551 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∧ (𝜓𝜃𝜂)))

Theorem3or6 1257 Analog of or4 721 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
(((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)))

Theoremmp3an1 1258 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜑    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜒) → 𝜃)

Theoremmp3an2 1259 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜓    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremmp3an3 1260 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremmp3an12 1261 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
𝜑    &   𝜓    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜒𝜃)

Theoremmp3an13 1262 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
𝜑    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜓𝜃)

Theoremmp3an23 1263 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
𝜓    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoremmp3an1i 1264 An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
𝜓    &   (𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑 → ((𝜒𝜃) → 𝜏))

Theoremmp3anl1 1265 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜑    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜓𝜒) ∧ 𝜃) → 𝜏)

Theoremmp3anl2 1266 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜓    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜒) ∧ 𝜃) → 𝜏)

Theoremmp3anl3 1267 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜒    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓) ∧ 𝜃) → 𝜏)

Theoremmp3anr1 1268 An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
𝜓    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜒𝜃)) → 𝜏)

Theoremmp3anr2 1269 An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
𝜒    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜃)) → 𝜏)

Theoremmp3anr3 1270 An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
𝜃    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒)) → 𝜏)

Theoremmp3an 1271 An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
𝜑    &   𝜓    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       𝜃

Theoremmpd3an3 1272 An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremmpd3an23 1273 An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoremmp3and 1274 A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑𝜏)

Theoremmp3an12i 1275 mp3an 1271 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   𝜓    &   (𝜒𝜃)    &   ((𝜑𝜓𝜃) → 𝜏)       (𝜒𝜏)

Theoremmp3an2i 1276 mp3an 1271 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   (𝜓𝜒)    &   (𝜓𝜃)    &   ((𝜑𝜒𝜃) → 𝜏)       (𝜓𝜏)

Theoremmp3an3an 1277 mp3an 1271 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   (𝜓𝜒)    &   (𝜃𝜏)    &   ((𝜑𝜒𝜏) → 𝜂)       ((𝜓𝜃) → 𝜂)

Theoremmp3an2ani 1278 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   (𝜓𝜒)    &   ((𝜓𝜃) → 𝜏)    &   ((𝜑𝜒𝜏) → 𝜂)       ((𝜓𝜃) → 𝜂)

Theorembiimp3a 1279 Infer implication from a logical equivalence. Similar to biimpa 290. (Contributed by NM, 4-Sep-2005.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembiimp3ar 1280 Infer implication from a logical equivalence. Similar to biimpar 291. (Contributed by NM, 2-Jan-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜃) → 𝜒)

Theorem3anandis 1281 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)
(((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)

Theorem3anandirs 1282 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.)
(((𝜑𝜃) ∧ (𝜓𝜃) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)

Theoremecased 1283 Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜓)

Theoremecase23d 1284 Variation of ecased 1283 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜓𝜒𝜃))       (𝜑𝜓)

1.2.13  True and false constants

1.2.13.1  Universal quantifier for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the universal quantifier is introduced here so that the soundness of definition df-tru 1290 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in axiom ax-5 1379 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 1295 may be adopted and this subsection moved down to the start of the subsection with wex 1424 below. However, the use of dftru2 1295 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

Syntaxwal 1285 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

1.2.13.2  Equality predicate for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1290 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in axiom ax-8 1438 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 1295 may be adopted and this subsection moved down to just above weq 1435 below. However, the use of dftru2 1295 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

Syntaxcv 1286 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 {𝑦𝑦𝑥} is a class by cab 2071. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2080, we can argue that the syntax "class 𝑥 " can be viewed as an abbreviation for "class {𝑦𝑦𝑥}". 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 1286 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1286 is intrinsically no different from any other class-building syntax such as cab 2071, cun 2986, or c0 3275.

For a general discussion of the theory of classes and the role of cv 1286, 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 1435 of predicate calculus from the wceq 1287 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

Syntaxwceq 1287 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 1435 of predicate calculus in terms of the wceq 1287 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 1435 or wceq 1287, 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 2078 for more information on the set theory usage of wceq 1287.)

wff 𝐴 = 𝐵

1.2.13.3  Define the true and false constants

Syntaxwtru 1288 is a wff.
wff

Theoremtrujust 1289 Soundness justification theorem for df-tru 1290. (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))

Definitiondf-tru 1290 Definition of the truth value "true", or "verum", denoted by . This is a tautology, as proved by tru 1291. 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 1291, and other proofs should depend on tru 1291 (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))

Theoremtru 1291 The truth value is provable. (Contributed by Anthony Hart, 13-Oct-2010.)

Syntaxwfal 1292 is a wff.
wff

Definitiondf-fal 1293 Definition of the truth value "false", or "falsum", denoted by . See also df-tru 1290. (Contributed by Anthony Hart, 22-Oct-2010.)
(⊥ ↔ ¬ ⊤)

Theoremfal 1294 The truth value is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
¬ ⊥

Theoremdftru2 1295 An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.)
(⊤ ↔ (𝜑𝜑))

Theoremtrud 1296 Eliminate as an antecedent. A proposition implied by is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
(⊤ → 𝜑)       𝜑

Theoremtbtru 1297 A proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
(𝜑 ↔ (𝜑 ↔ ⊤))

Theoremnbfal 1298 The negation of a proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
𝜑 ↔ (𝜑 ↔ ⊥))

Theorembitru 1299 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
𝜑       (𝜑 ↔ ⊤)

Theorembifal 1300 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
¬ 𝜑       (𝜑 ↔ ⊥)

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