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Theorem List for Metamath Proof Explorer - 1301-1400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimp132 1301 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂𝜁) → 𝜓)
 
Theoremsimp133 1302 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂𝜁) → 𝜒)
 
Theoremsimp211 1303 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜑)
 
Theoremsimp212 1304 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜓)
 
Theoremsimp213 1305 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜒)
 
Theoremsimp221 1306 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜑)
 
Theoremsimp222 1307 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜓)
 
Theoremsimp223 1308 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜒)
 
Theoremsimp231 1309 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜑)
 
Theoremsimp232 1310 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜓)
 
Theoremsimp233 1311 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜒)
 
Theoremsimp311 1312 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜑)
 
Theoremsimp312 1313 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜓)
 
Theoremsimp313 1314 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜒)
 
Theoremsimp321 1315 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜑)
 
Theoremsimp322 1316 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜓)
 
Theoremsimp323 1317 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜒)
 
Theoremsimp331 1318 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜑)
 
Theoremsimp332 1319 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜓)
 
Theoremsimp333 1320 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜒)
 
Theorem3anibar 1321 Remove a hypothesis from the second member of a biimplication. (Contributed by FL, 22-Jul-2008.)
((𝜑𝜓𝜒) → (𝜃 ↔ (𝜒𝜏)))       ((𝜑𝜓𝜒) → (𝜃𝜏))
 
Theorem3mix1 1322 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜑𝜓𝜒))
 
Theorem3mix2 1323 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜓𝜑𝜒))
 
Theorem3mix3 1324 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜓𝜒𝜑))
 
Theorem3mix1i 1325 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜑𝜓𝜒)
 
Theorem3mix2i 1326 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜓𝜑𝜒)
 
Theorem3mix3i 1327 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜓𝜒𝜑)
 
Theorem3mix1d 1328 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒𝜃))
 
Theorem3mix2d 1329 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓𝜃))
 
Theorem3mix3d 1330 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜒𝜃𝜓))
 
Theorem3pm3.2i 1331 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
𝜑    &   𝜓    &   𝜒       (𝜑𝜓𝜒)
 
Theorempm3.2an3 1332 Version of pm3.2 470 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Kyle Wyonch, 24-Apr-2021.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
(𝜑 → (𝜓 → (𝜒 → (𝜑𝜓𝜒))))
 
Theoremmpbir3an 1333 Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.)
𝜓    &   𝜒    &   𝜃    &   (𝜑 ↔ (𝜓𝜒𝜃))       𝜑
 
Theoremmpbir3and 1334 Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)
(𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃𝜏)))       (𝜑𝜓)
 
Theoremsyl3anbrc 1335 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜏 ↔ (𝜓𝜒𝜃))       (𝜑𝜏)
 
Theoremsyl21anbrc 1336 Syllogism inference. (Contributed by Peter Mazsa, 18-Sep-2022.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜏 ↔ ((𝜓𝜒) ∧ 𝜃))       (𝜑𝜏)
 
Theorem3imp3i2an 1337 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)    &   ((𝜑𝜒) → 𝜏)    &   ((𝜃𝜏) → 𝜂)       ((𝜑𝜓𝜒) → 𝜂)
 
Theoremex3 1338 Apply ex 413 to a hypothesis with a 3-right-nested conjunction antecedent, with the antecedent of the assertion being a triple conjunction rather than a 2-right-nested conjunction. (Contributed by Alan Sare, 22-Apr-2018.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((𝜑𝜓𝜒) → (𝜃𝜏))
 
Theorem3imp1 1339 Importation to left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)
 
Theorem3impd 1340 Importation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → ((𝜓𝜒𝜃) → 𝜏))
 
Theorem3imp2 1341 Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)
 
Theorem3impdi 1342 Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impdir 1343 Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜓)) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)
 
Theorem3exp1 1344 Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theorem3expd 1345 Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theorem3exp2 1346 Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp5o 1347 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
((𝜑𝜓𝜒) → ((𝜃𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp516 1348 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp520 1349 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theorem3impexp 1350 Version of impexp 451 for a triple conjunction. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
 
Theorem3an1rs 1351 Swap conjuncts. (Contributed by NM, 16-Dec-2007.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓𝜃) ∧ 𝜒) → 𝜏)
 
Theorem3anassrs 1352 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theoremad5ant245 1353 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)
 
Theoremad5ant234 1354 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)
 
Theoremad5ant235 1355 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)
 
Theoremad5ant123 1356 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜃)
 
Theoremad5ant124 1357 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜓) ∧ 𝜏) ∧ 𝜒) ∧ 𝜂) → 𝜃)
 
Theoremad5ant125 1358 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜓) ∧ 𝜏) ∧ 𝜂) ∧ 𝜒) → 𝜃)
 
Theoremad5ant134 1359 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜏) ∧ 𝜓) ∧ 𝜒) ∧ 𝜂) → 𝜃)
 
Theoremad5ant135 1360 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜏) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃)
 
Theoremad5ant145 1361 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)
((𝜑𝜓𝜒) → 𝜃)       (((((𝜑𝜏) ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) → 𝜃)
 
Theoremad5ant2345 1362 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (((((𝜂𝜑) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theoremsyl3anc 1363 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       (𝜑𝜏)
 
Theoremsyl13anc 1364 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((𝜓 ∧ (𝜒𝜃𝜏)) → 𝜂)       (𝜑𝜂)
 
Theoremsyl31anc 1365 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (𝜑𝜂)
 
Theoremsyl112anc 1366 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((𝜓𝜒 ∧ (𝜃𝜏)) → 𝜂)       (𝜑𝜂)
 
Theoremsyl121anc 1367 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((𝜓 ∧ (𝜒𝜃) ∧ 𝜏) → 𝜂)       (𝜑𝜂)
 
Theoremsyl211anc 1368 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (((𝜓𝜒) ∧ 𝜃𝜏) → 𝜂)       (𝜑𝜂)
 
Theoremsyl23anc 1369 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂)) → 𝜁)       (𝜑𝜁)
 
Theoremsyl32anc 1370 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)
 
Theoremsyl122anc 1371 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)
 
Theoremsyl212anc 1372 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)
 
Theoremsyl221anc 1373 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ 𝜂) → 𝜁)       (𝜑𝜁)
 
Theoremsyl113anc 1374 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓𝜒 ∧ (𝜃𝜏𝜂)) → 𝜁)       (𝜑𝜁)
 
Theoremsyl131anc 1375 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ 𝜂) → 𝜁)       (𝜑𝜁)
 
Theoremsyl311anc 1376 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒𝜃) ∧ 𝜏𝜂) → 𝜁)       (𝜑𝜁)
 
Theoremsyl33anc 1377 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl222anc 1378 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl123anc 1379 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl132anc 1380 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl213anc 1381 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl231anc 1382 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ 𝜁) → 𝜎)       (𝜑𝜎)
 
Theoremsyl312anc 1383 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)
 
Theoremsyl321anc 1384 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ 𝜁) → 𝜎)       (𝜑𝜎)
 
Theoremsyl133anc 1385 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)
 
Theoremsyl313anc 1386 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)
 
Theoremsyl331anc 1387 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)       (𝜑𝜌)
 
Theoremsyl223anc 1388 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)
 
Theoremsyl232anc 1389 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)       (𝜑𝜌)
 
Theoremsyl322anc 1390 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)       (𝜑𝜌)
 
Theoremsyl233anc 1391 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)       (𝜑𝜇)
 
Theoremsyl323anc 1392 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)       (𝜑𝜇)
 
Theoremsyl332anc 1393 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌)) → 𝜇)       (𝜑𝜇)
 
Theoremsyl333anc 1394 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)
 
Theoremsyl3an1b 1395 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜓)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)
 
Theoremsyl3an2b 1396 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜒)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)
 
Theoremsyl3an3b 1397 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)
 
Theoremsyl3an1br 1398 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜓𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)
 
Theoremsyl3an2br 1399 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜒𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)
 
Theoremsyl3an3br 1400 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜃𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)
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