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Theorem List for Metamath Proof Explorer - 1301-1400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem3an1rsOLD 1301 Obsolete version of 3an1rs 1312 as of 14-Apr-2022. (Contributed by NM, 16-Dec-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓𝜃) ∧ 𝜒) → 𝜏)

Theorem3imp1 1302 Importation to left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)

Theorem3impd 1303 Importation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → ((𝜓𝜒𝜃) → 𝜏))

Theorem3imp2 1304 Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)

Theorem3exp1 1305 Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theorem3expd 1306 Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theorem3exp2 1307 Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp5o 1308 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
((𝜑𝜓𝜒) → ((𝜃𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp516 1309 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp520 1310 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theorem3impexp 1311 Version of impexp 461 for a triple conjunction. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))

Theorem3an1rs 1312 Swap conjuncts. (Contributed by NM, 16-Dec-2007.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓𝜃) ∧ 𝜒) → 𝜏)

Theorem3anassrs 1313 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem3an4anass 1314 Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
(((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))

Theoremad4ant13 1315 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       ((((𝜑𝜃) ∧ 𝜓) ∧ 𝜏) → 𝜒)

Theoremad4ant13OLD 1316 Obsolete proof of ad4ant13 1315 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       ((((𝜑𝜃) ∧ 𝜓) ∧ 𝜏) → 𝜒)

Theoremad4ant14 1317 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       ((((𝜑𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜒)

Theoremad4ant14OLD 1318 Obsolete version of ad4ant14 1317 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       ((((𝜑𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜒)

Theoremad4ant123 1319 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)

Theoremad4ant123OLD 1320 Obsolete version of ad4ant123 1319 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜃)

Theoremad4ant124 1321 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)

Theoremad4ant124OLD 1322 Obsolete version of ad4ant124 1321 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜓) ∧ 𝜏) ∧ 𝜒) → 𝜃)

Theoremad4ant134 1323 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)

Theoremad4ant134OLD 1324 Obsolete version of ad4ant134 1323 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Theoremad4ant23 1325 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       ((((𝜃𝜑) ∧ 𝜓) ∧ 𝜏) → 𝜒)

Theoremad4ant23OLD 1326 Obsolete version of ad4ant23 1325 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       ((((𝜃𝜑) ∧ 𝜓) ∧ 𝜏) → 𝜒)

Theoremad4ant24 1327 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       ((((𝜃𝜑) ∧ 𝜏) ∧ 𝜓) → 𝜒)

Theoremad4ant24OLD 1328 Obsolete version of ad4ant24 1327 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       ((((𝜃𝜑) ∧ 𝜏) ∧ 𝜓) → 𝜒)

Theoremad4ant234 1329 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)

Theoremad4ant234OLD 1330 Obsolete version of ad4ant234 1329 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜏𝜑) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Theoremad5ant12 1331 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜓) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜒)

Theoremad5ant13 1332 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) → 𝜒)

Theoremad5ant13OLD 1333 Obsolete version of ad5ant13 1332 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) → 𝜒)

Theoremad5ant14 1334 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)

Theoremad5ant14OLD 1335 Obsolete version of ad5ant14 1334 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)

Theoremad5ant15 1336 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)

Theoremad5ant15OLD 1337 Obsolete proof of ad5ant15 1336 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       (((((𝜑𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)

Theoremad5ant23 1338 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) → 𝜒)

Theoremad5ant23OLD 1339 Obsolete version of ad5ant23 1338 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) → 𝜒)

Theoremad5ant24 1340 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)

Theoremad5ant24OLD 1341 Obsolete version of ad5ant24 1340 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)

Theoremad5ant25 1342 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)

Theoremad5ant25OLD 1343 Obsolete version of ad5ant25 1342 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       (((((𝜃𝜑) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)

Theoremad5ant245 1344 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜏𝜑) ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Theoremad5ant245OLD 1345 Obsolete version of ad5ant245 1344 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜏𝜑) ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Theoremad5ant234 1346 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) ∧ 𝜂) → 𝜃)

Theoremad5ant234OLD 1347 Obsolete version of ad5ant234 1346 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜏𝜑) ∧ 𝜓) ∧ 𝜒) ∧ 𝜂) → 𝜃)

Theoremad5ant235 1348 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜏𝜑) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃)

Theoremad5ant235OLD 1349 Obsolete version of ad5ant235 1348 as of 14-Apr-2022. (Contributed by Alan Sare, 17-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜏𝜑) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃)

Theoremad5ant123 1350 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜃)

Theoremad5ant124 1351 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜓) ∧ 𝜏) ∧ 𝜒) ∧ 𝜂) → 𝜃)

Theoremad5ant125 1352 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜓) ∧ 𝜏) ∧ 𝜂) ∧ 𝜒) → 𝜃)

Theoremad5ant134 1353 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜏) ∧ 𝜓) ∧ 𝜒) ∧ 𝜂) → 𝜃)

Theoremad5ant135 1354 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜏) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃)

Theoremad5ant145 1355 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜏) ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Theoremad5ant1345 1356 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (((((𝜑𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theoremad5ant2345 1357 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (((((𝜂𝜑) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem3adant1l 1358 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜏𝜑) ∧ 𝜓𝜒) → 𝜃)

Theorem3adant1r 1359 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜑𝜏) ∧ 𝜓𝜒) → 𝜃)

Theorem3adant2l 1360 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜏𝜓) ∧ 𝜒) → 𝜃)

Theorem3adant2r 1361 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏) ∧ 𝜒) → 𝜃)

Theorem3adant3l 1362 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓 ∧ (𝜏𝜒)) → 𝜃)

Theorem3adant3r 1363 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓 ∧ (𝜒𝜏)) → 𝜃)

Theoremsyl12anc 1364 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       (𝜑𝜏)

Theoremsyl21anc 1365 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (𝜑𝜏)

Theoremsyl3anc 1366 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       (𝜑𝜏)

Theoremsyl22anc 1367 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (((𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑𝜂)

Theoremsyl13anc 1368 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((𝜓 ∧ (𝜒𝜃𝜏)) → 𝜂)       (𝜑𝜂)

Theoremsyl31anc 1369 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (𝜑𝜂)

Theoremsyl112anc 1370 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((𝜓𝜒 ∧ (𝜃𝜏)) → 𝜂)       (𝜑𝜂)

Theoremsyl121anc 1371 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((𝜓 ∧ (𝜒𝜃) ∧ 𝜏) → 𝜂)       (𝜑𝜂)

Theoremsyl211anc 1372 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (((𝜓𝜒) ∧ 𝜃𝜏) → 𝜂)       (𝜑𝜂)

Theoremsyl23anc 1373 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂)) → 𝜁)       (𝜑𝜁)

Theoremsyl32anc 1374 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)

Theoremsyl122anc 1375 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)

Theoremsyl212anc 1376 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)

Theoremsyl221anc 1377 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ 𝜂) → 𝜁)       (𝜑𝜁)

Theoremsyl113anc 1378 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓𝜒 ∧ (𝜃𝜏𝜂)) → 𝜁)       (𝜑𝜁)

Theoremsyl131anc 1379 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ 𝜂) → 𝜁)       (𝜑𝜁)

Theoremsyl311anc 1380 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒𝜃) ∧ 𝜏𝜂) → 𝜁)       (𝜑𝜁)

Theoremsyl33anc 1381 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl222anc 1382 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl123anc 1383 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl132anc 1384 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl213anc 1385 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl231anc 1386 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ 𝜁) → 𝜎)       (𝜑𝜎)

Theoremsyl312anc 1387 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl321anc 1388 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ 𝜁) → 𝜎)       (𝜑𝜎)

Theoremsyl133anc 1389 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)

Theoremsyl313anc 1390 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)

Theoremsyl331anc 1391 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)       (𝜑𝜌)

Theoremsyl223anc 1392 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)

Theoremsyl232anc 1393 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)       (𝜑𝜌)

Theoremsyl322anc 1394 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)       (𝜑𝜌)

Theoremsyl233anc 1395 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)       (𝜑𝜇)

Theoremsyl323anc 1396 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)       (𝜑𝜇)

Theoremsyl332anc 1397 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌)) → 𝜇)       (𝜑𝜇)

Theoremsyl333anc 1398 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 1399 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜓)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)

Theoremsyl3an2 1400 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜒)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)

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