mmtheorems15 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1401-1500 - Page 15 of 195

Type	Label	Description
Statement
Theorem	syl313anc 1401	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (((ps /\ ch /\ th) /\ ta /\ (et /\ ze /\ si)) -> rh)   =>   |- (ph -> rh)
Theorem	syl331anc 1402	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et /\ ze) /\ si) -> rh)   =>   |- (ph -> rh)
Theorem	syl223anc 1403	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (((ps /\ ch) /\ (th /\ ta) /\ (et /\ ze /\ si)) -> rh)   =>   |- (ph -> rh)
Theorem	syl232anc 1404	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (((ps /\ ch) /\ (th /\ ta /\ et) /\ (ze /\ si)) -> rh)   =>   |- (ph -> rh)
Theorem	syl322anc 1405	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et) /\ (ze /\ si)) -> rh)   =>   |- (ph -> rh)
Theorem	syl233anc 1406	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (ph -> rh)   &   |- (((ps /\ ch) /\ (th /\ ta /\ et) /\ (ze /\ si /\ rh)) -> mu)   =>   |- (ph -> mu)
Theorem	syl323anc 1407	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (ph -> rh)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et) /\ (ze /\ si /\ rh)) -> mu)   =>   |- (ph -> mu)
Theorem	syl332anc 1408	Syllogism combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (ph -> rh)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et /\ ze) /\ (si /\ rh)) -> mu)   =>   |- (ph -> mu)
Theorem	syl333anc 1409	A syllogism inference combined with contraction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   &   |- (ph -> ta)   &   |- (ph -> et)   &   |- (ph -> ze)   &   |- (ph -> si)   &   |- (ph -> rh)   &   |- (ph -> mu)   &   |- (((ps /\ ch /\ th) /\ (ta /\ et /\ ze) /\ (si /\ rh /\ mu)) -> la)   =>   |- (ph -> la)
Theorem	syl3an1 1410	A syllogism inference.
|- (ph -> ps)   &   |- ((ps /\ ch /\ th) -> ta)   =>   |- ((ph /\ ch /\ th) -> ta)
Theorem	syl3an2 1411	A syllogism inference.
|- (ph -> ch)   &   |- ((ps /\ ch /\ th) -> ta)   =>   |- ((ps /\ ph /\ th) -> ta)
Theorem	syl3an3 1412	A syllogism inference.
|- (ph -> th)   &   |- ((ps /\ ch /\ th) -> ta)   =>   |- ((ps /\ ch /\ ph) -> ta)
Theorem	syl3an1b 1413	A syllogism inference.
|- (ph <-> ps)   &   |- ((ps /\ ch /\ th) -> ta)   =>   |- ((ph /\ ch /\ th) -> ta)
Theorem	syl3an2b 1414	A syllogism inference.
|- (ph <-> ch)   &   |- ((ps /\ ch /\ th) -> ta)   =>   |- ((ps /\ ph /\ th) -> ta)
Theorem	syl3an3b 1415	A syllogism inference.
|- (ph <-> th)   &   |- ((ps /\ ch /\ th) -> ta)   =>   |- ((ps /\ ch /\ ph) -> ta)
Theorem	syl3an1br 1416	A syllogism inference.
|- (ps <-> ph)   &   |- ((ps /\ ch /\ th) -> ta)   =>   |- ((ph /\ ch /\ th) -> ta)
Theorem	syl3an2br 1417	A syllogism inference.
|- (ch <-> ph)   &   |- ((ps /\ ch /\ th) -> ta)   =>   |- ((ps /\ ph /\ th) -> ta)
Theorem	syl3an3br 1418	A syllogism inference.
|- (th <-> ph)   &   |- ((ps /\ ch /\ th) -> ta)   =>   |- ((ps /\ ch /\ ph) -> ta)
Theorem	syl3an 1419	A triple syllogism inference.
|- (ph -> ps)   &   |- (ch -> th)   &   |- (ta -> et)   &   |- ((ps /\ th /\ et) -> ze)   =>   |- ((ph /\ ch /\ ta) -> ze)
Theorem	syl3anb 1420	A triple syllogism inference.
|- (ph <-> ps)   &   |- (ch <-> th)   &   |- (ta <-> et)   &   |- ((ps /\ th /\ et) -> ze)   =>   |- ((ph /\ ch /\ ta) -> ze)
Theorem	syl3anbr 1421	A triple syllogism inference.
|- (ps <-> ph)   &   |- (th <-> ch)   &   |- (et <-> ta)   &   |- ((ps /\ th /\ et) -> ze)   =>   |- ((ph /\ ch /\ ta) -> ze)
Theorem	syld3an3 1422	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- ((ph /\ ps /\ th) -> ta)   =>   |- ((ph /\ ps /\ ch) -> ta)
Theorem	syld3an1 1423	A syllogism inference.
|- ((ch /\ ps /\ th) -> ph)   &   |- ((ph /\ ps /\ th) -> ta)   =>   |- ((ch /\ ps /\ th) -> ta)
Theorem	syld3an2 1424	A syllogism inference.
|- ((ph /\ ch /\ th) -> ps)   &   |- ((ph /\ ps /\ th) -> ta)   =>   |- ((ph /\ ch /\ th) -> ta)
Theorem	syl3an1OLD 1425	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ph)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syl3an2OLD 1426	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ps)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3an3OLD 1427	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ch)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syl3an1bOLD 1428	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ph)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syl3an2bOLD 1429	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ps)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3an3bOLD 1430	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ch)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syl3an1brOLD 1431	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ph <-> ta)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syl3an2brOLD 1432	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ps <-> ta)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3an3brOLD 1433	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ch <-> ta)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syl3anOLD 1434	A triple syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ph)   &   |- (et -> ps)   &   |- (ze -> ch)   =>   |- ((ta /\ et /\ ze) -> th)
Theorem	syl3anbOLD 1435	A triple syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ph)   &   |- (et <-> ps)   &   |- (ze <-> ch)   =>   |- ((ta /\ et /\ ze) -> th)
Theorem	syl3anbrOLD 1436	A triple syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ph <-> ta)   &   |- (ps <-> et)   &   |- (ch <-> ze)   =>   |- ((ta /\ et /\ ze) -> th)
Theorem	syld3an3OLD 1437	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- ((ph /\ ps /\ ta) -> ch)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syld3an1OLD 1438	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- ((ta /\ ps /\ ch) -> ph)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syld3an2OLD 1439	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- ((ph /\ ta /\ ch) -> ps)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3anl1 1440	A syllogism inference.
|- (ph -> ps)   &   |- (((ps /\ ch /\ th) /\ ta) -> et)   =>   |- (((ph /\ ch /\ th) /\ ta) -> et)
Theorem	syl3anl2 1441	A syllogism inference.
|- (ph -> ch)   &   |- (((ps /\ ch /\ th) /\ ta) -> et)   =>   |- (((ps /\ ph /\ th) /\ ta) -> et)
Theorem	syl3anl3 1442	A syllogism inference.
|- (ph -> th)   &   |- (((ps /\ ch /\ th) /\ ta) -> et)   =>   |- (((ps /\ ch /\ ph) /\ ta) -> et)
Theorem	syl3anl 1443	A triple syllogism inference.
|- (ph -> ps)   &   |- (ch -> th)   &   |- (ta -> et)   &   |- (((ps /\ th /\ et) /\ ze) -> si)   =>   |- (((ph /\ ch /\ ta) /\ ze) -> si)
Theorem	syl3anl1OLD 1444	A syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ph)   =>   |- (((et /\ ps /\ ch) /\ th) -> ta)
Theorem	syl3anl2OLD 1445	A syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ps)   =>   |- (((ph /\ et /\ ch) /\ th) -> ta)
Theorem	syl3anl3OLD 1446	A syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ch)   =>   |- (((ph /\ ps /\ et) /\ th) -> ta)
Theorem	syl3anlOLD 1447	A triple syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ph)   &   |- (ze -> ps)   &   |- (si -> ch)   =>   |- (((et /\ ze /\ si) /\ th) -> ta)
Theorem	syl3anr1 1448	A syllogism inference.
|- (ph -> ps)   &   |- ((ch /\ (ps /\ th /\ ta)) -> et)   =>   |- ((ch /\ (ph /\ th /\ ta)) -> et)
Theorem	syl3anr2 1449	A syllogism inference.
|- (ph -> th)   &   |- ((ch /\ (ps /\ th /\ ta)) -> et)   =>   |- ((ch /\ (ps /\ ph /\ ta)) -> et)
Theorem	syl3anr3 1450	A syllogism inference.
|- (ph -> ta)   &   |- ((ch /\ (ps /\ th /\ ta)) -> et)   =>   |- ((ch /\ (ps /\ th /\ ph)) -> et)
Theorem	syl3anr1OLD 1451	A syllogism inference.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   &   |- (et -> ps)   =>   |- ((ph /\ (et /\ ch /\ th)) -> ta)
Theorem	syl3anr2OLD 1452	A syllogism inference.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   &   |- (et -> ch)   =>   |- ((ph /\ (ps /\ et /\ th)) -> ta)
Theorem	syl3anr3OLD 1453	A syllogism inference.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   &   |- (et -> th)   =>   |- ((ph /\ (ps /\ ch /\ et)) -> ta)
Theorem	3impdi 1454	Importation inference (undistribute conjunction).
|- (((ph /\ ps) /\ (ph /\ ch)) -> th)   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impdir 1455	Importation inference (undistribute conjunction).
|- (((ph /\ ps) /\ (ch /\ ps)) -> th)   =>   |- ((ph /\ ch /\ ps) -> th)
Theorem	3anidm12 1456	Inference from idempotent law for conjunction.
|- ((ph /\ ph /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	3anidm13 1457	Inference from idempotent law for conjunction.
|- ((ph /\ ps /\ ph) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	3anidm23 1458	Inference from idempotent law for conjunction.
|- ((ph /\ ps /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	3ori 1459	Infer implication from triple disjunction.
|- (ph \/ ps \/ ch)   =>   |- ((-. ph /\ -. ps) -> ch)
Theorem	3jao 1460	Disjunction of 3 antecedents.
|- (((ph -> ps) /\ (ch -> ps) /\ (th -> ps)) -> ((ph \/ ch \/ th) -> ps))
Theorem	3jaob 1461	Disjunction of 3 antecedents.
|- (((ph \/ ch \/ th) -> ps) <-> ((ph -> ps) /\ (ch -> ps) /\ (th -> ps)))
Theorem	3jaoi 1462	Disjunction of 3 antecedents (inference).
|- (ph -> ps)   &   |- (ch -> ps)   &   |- (th -> ps)   =>   |- ((ph \/ ch \/ th) -> ps)
Theorem	3jaod 1463	Disjunction of 3 antecedents (deduction).
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ch))   &   |- (ph -> (ta -> ch))   =>   |- (ph -> ((ps \/ th \/ ta) -> ch))
Theorem	3jaoian 1464	Disjunction of 3 antecedents (inference).
|- ((ph /\ ps) -> ch)   &   |- ((th /\ ps) -> ch)   &   |- ((ta /\ ps) -> ch)   =>   |- (((ph \/ th \/ ta) /\ ps) -> ch)
Theorem	3jaodan 1465	Disjunction of 3 antecedents (deduction).
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ th) -> ch)   &   |- ((ph /\ ta) -> ch)   =>   |- ((ph /\ (ps \/ th \/ ta)) -> ch)
Theorem	3jaao 1466	Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (The proof was shortened by Andrew Salmon, 13-May-2011.)
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   &   |- (et -> (ze -> ch))   =>   |- ((ph /\ th /\ et) -> ((ps \/ ta \/ ze) -> ch))
Theorem	syl3an9b 1467	Nested syllogism inference conjoining 3 dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta))   &   |- (et -> (ta <-> ze))   =>   |- ((ph /\ th /\ et) -> (ps <-> ze))
Theorem	3orbi123d 1468	Deduction joining 3 equivalences to form equivalence of disjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   &   |- (ph -> (et <-> ze))   =>   |- (ph -> ((ps \/ th \/ et) <-> (ch \/ ta \/ ze)))
Theorem	3anbi123d 1469	Deduction joining 3 equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   &   |- (ph -> (et <-> ze))   =>   |- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ ze)))
Theorem	3anbi12d 1470	Deduction conjoining and adding a conjunct to equivalences.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ et)))
Theorem	3anbi13d 1471	Deduction conjoining and adding a conjunct to equivalences.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps /\ et /\ th) <-> (ch /\ et /\ ta)))
Theorem	3anbi23d 1472	Deduction conjoining and adding a conjunct to equivalences.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((et /\ ps /\ th) <-> (et /\ ch /\ ta)))
Theorem	3anbi1d 1473	Deduction adding conjuncts to an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps /\ th /\ ta) <-> (ch /\ th /\ ta)))
Theorem	3anbi2d 1474	Deduction adding conjuncts to an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th /\ ps /\ ta) <-> (th /\ ch /\ ta)))
Theorem	3anbi3d 1475	Deduction adding conjuncts to an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th /\ ta /\ ps) <-> (th /\ ta /\ ch)))
Theorem	3anim123d 1476	Deduction joining 3 implications to form implication of conjunctions.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta))   &   |- (ph -> (et -> ze))   =>   |- (ph -> ((ps /\ th /\ et) -> (ch /\ ta /\ ze)))
Theorem	3orim123d 1477	Deduction joining 3 implications to form implication of disjunctions.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta))   &   |- (ph -> (et -> ze))   =>   |- (ph -> ((ps \/ th \/ et) -> (ch \/ ta \/ ze)))
Theorem	an6 1478	Rearrangement of 6 conjuncts.
|- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> ((ph /\ th) /\ (ps /\ ta) /\ (ch /\ et)))
Theorem	3an6 1479	Analogue of an4 939 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (((ph /\ ps) /\ (ch /\ th) /\ (ta /\ et)) <-> ((ph /\ ch /\ ta) /\ (ps /\ th /\ et)))
Theorem	3or6 1480	Analogue of or4 586 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
|- (((ph \/ ps) \/ (ch \/ th) \/ (ta \/ et)) <-> ((ph \/ ch \/ ta) \/ (ps \/ th \/ et)))
Theorem	mp3an1 1481	An inference based on modus ponens.
|- ph   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ch) -> th)
Theorem	mp3an2 1482	An inference based on modus ponens.
|- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mp3an3 1483	An inference based on modus ponens.
|- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mp3an12 1484	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ch -> th)
Theorem	mp3an13 1485	An inference based on modus ponens.
|- ph   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ps -> th)
Theorem	mp3an23 1486	An inference based on modus ponens.
|- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	mp3an1i 1487	An inference based on modus ponens.
|- ps   &   |- (ph -> ((ps /\ ch /\ th) -> ta))   =>   |- (ph -> ((ch /\ th) -> ta))
Theorem	mp3anl1 1488	An inference based on modus ponens.
|- ph   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ps /\ ch) /\ th) -> ta)
Theorem	mp3anl2 1489	An inference based on modus ponens.
|- ps   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ch) /\ th) -> ta)
Theorem	mp3anl3 1490	An inference based on modus ponens.
|- ch   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ps) /\ th) -> ta)
Theorem	mp3anr1 1491	An inference based on modus ponens.
|- ps   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ch /\ th)) -> ta)
Theorem	mp3anr2 1492	An inference based on modus ponens.
|- ch   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ th)) -> ta)
Theorem	mp3anr3 1493	An inference based on modus ponens.
|- th   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch)) -> ta)
Theorem	mp3an 1494	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th
Theorem	mpd3an3 1495	An inference based on modus ponens.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpd3an23 1496	An inference based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	biimp3a 1497	Infer implication from a logical equivalence. Similar to biimpa 711.
|- ((ph /\ ps) -> (ch <-> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	biimp3ar 1498	Infer implication from a logical equivalence. Similar to biimpar 712.
|- ((ph /\ ps) -> (ch <-> th))   =>   |- ((ph /\ ps /\ th) -> ch)
Theorem	3anandis 1499	Inference that undistributes a triple conjunction in the antecedent.
|- (((ph /\ ps) /\ (ph /\ ch) /\ (ph /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch /\ th)) -> ta)
Theorem	3anandirs 1500	Inference that undistributes a triple conjunction in the antecedent.
|- (((ph /\ th) /\ (ps /\ th) /\ (ch /\ th)) -> ta)   =>   |- (((ph /\ ps /\ ch) /\ th) -> ta)