mmtheorems5 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12229

Statement List for Metamath Proof Explorer - 401-500 - Page 5 of 123

Type	Label	Description
Statement
Theorem	adantrlr 401	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ((ps /\ ta) /\ ch)) -> th)
Theorem	adantrrl 402	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ (ta /\ ch))) -> th)
Theorem	adantrrr 403	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ (ch /\ ta))) -> th)
Theorem	ad2antrr 404	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- (((ph /\ ch) /\ th) -> ps)
Theorem	ad2antlr 405	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- (((ch /\ ph) /\ th) -> ps)
Theorem	ad2antrl 406	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- ((ch /\ (ph /\ th)) -> ps)
Theorem	ad2antll 407	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- ((ch /\ (th /\ ph)) -> ps)
Theorem	ad2ant2l 408	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ (ta /\ ps)) -> ch)
Theorem	ad2ant2r 409	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ (ps /\ ta)) -> ch)
Theorem	ad2ant2lr 410	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ (ps /\ ta)) -> ch)
Theorem	ad2ant2rl 411	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ (ta /\ ps)) -> ch)
Theorem	simpll 412	Simplification of a conjunction.
|- (((ph /\ ps) /\ ch) -> ph)
Theorem	simplr 413	Simplification of a conjunction.
|- (((ph /\ ps) /\ ch) -> ps)
Theorem	simprl 414	Simplification of a conjunction.
|- ((ph /\ (ps /\ ch)) -> ps)
Theorem	simprr 415	Simplification of a conjunction.
|- ((ph /\ (ps /\ ch)) -> ch)
Theorem	biimpa 416	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ps) -> ch)
Theorem	biimpar 417	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ch) -> ps)
Theorem	biimpac 418	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ps /\ ph) -> ch)
Theorem	biimparc 419	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ch /\ ph) -> ps)
Theorem	pm3.26bda 420	Deduction eliminating a conjunct.
|- (ph -> (ps <-> (ch /\ th)))   =>   |- ((ph /\ ps) -> ch)
Theorem	pm3.27bda 421	Deduction eliminating a conjunct.
|- (ph -> (ps <-> (ch /\ th)))   =>   |- ((ph /\ ps) -> th)
Theorem	jaob 422	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121.
|- (((ph \/ ch) -> ps) <-> ((ph -> ps) /\ (ch -> ps)))
Theorem	pm4.77 423	Theorem *4.77 of [WhiteheadRussell] p. 121.
|- (((ps -> ph) /\ (ch -> ph)) <-> ((ps \/ ch) -> ph))
Theorem	jaod 424	Deduction disjoining the antecedents of two implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ch))   =>   |- (ph -> ((ps \/ th) -> ch))
Theorem	jaoian 425	Inference disjoining the antecedents of two implications.
|- ((ph /\ ps) -> ch)   &   |- ((th /\ ps) -> ch)   =>   |- (((ph \/ th) /\ ps) -> ch)
Theorem	jaodan 426	Deduction disjoining the antecedents of two implications.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ th) -> ch)   =>   |- ((ph /\ (ps \/ th)) -> ch)
Theorem	jaao 427	Inference conjoining and disjoining the antecedents of two implications.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   =>   |- ((ph /\ th) -> ((ps \/ ta) -> ch))
Theorem	jaoa 428	Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   =>   |- ((ph \/ th) -> ((ps /\ ta) -> ch))
Theorem	pm2.63 429	Theorem *2.63 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> ((-. ph \/ ps) -> ps))
Theorem	pm2.64 430	Theorem *2.64 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> ((ph \/ -. ps) -> ph))
Theorem	pm3.44 431	Theorem *3.44 of [WhiteheadRussell] p. 113.
|- (((ps -> ph) /\ (ch -> ph)) -> ((ps \/ ch) -> ph))
Theorem	pm4.43 432	Theorem *4.43 of [WhiteheadRussell] p. 119.
|- (ph <-> ((ph \/ ps) /\ (ph \/ -. ps)))
Theorem	anidm 433	Idempotent law for conjunction.
|- ((ph /\ ph) <-> ph)
Theorem	pm4.24 434	Theorem *4.24 of [WhiteheadRussell] p. 117.
|- (ph <-> (ph /\ ph))
Theorem	anidms 435	Inference from idempotent law for conjunction.
|- ((ph /\ ph) -> ps)   =>   |- (ph -> ps)
Theorem	anidmdbi 436	Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
|- ((ph -> (ps /\ ps)) <-> (ph -> ps))
Theorem	ancom 437	Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118.
|- ((ph /\ ps) <-> (ps /\ ph))
Theorem	ancoms 438	Inference commuting conjunction in antecedent. Notational convention: We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 10) -type inference in a proof.
|- ((ph /\ ps) -> ch)   =>   |- ((ps /\ ph) -> ch)
Theorem	ancomsd 439	Deduction commuting conjunction in antecedent.
|- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> ((ch /\ ps) -> th))
Theorem	pm3.22 440	Theorem *3.22 of [WhiteheadRussell] p. 111.
|- ((ph /\ ps) -> (ps /\ ph))
Theorem	anass 441	Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118.
|- (((ph /\ ps) /\ ch) <-> (ph /\ (ps /\ ch)))
Theorem	anasss 442	Associative law for conjunction applied to antecedent (eliminates syllogism).
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch)) -> th)
Theorem	anassrs 443	Associative law for conjunction applied to antecedent (eliminates syllogism).
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- (((ph /\ ps) /\ ch) -> th)
Theorem	imdistan 444	Distribution of implication with conjunction.
|- ((ph -> (ps -> ch)) <-> ((ph /\ ps) -> (ph /\ ch)))
Theorem	imdistani 445	Distribution of implication with conjunction.
|- (ph -> (ps -> ch))   =>   |- ((ph /\ ps) -> (ph /\ ch))
Theorem	imdistanri 446	Distribution of implication with conjunction.
|- (ph -> (ps -> ch))   =>   |- ((ps /\ ph) -> (ch /\ ph))
Theorem	imdistand 447	Distribution of implication with conjunction (deduction rule).
|- (ph -> (ps -> (ch -> th)))   =>   |- (ph -> ((ps /\ ch) -> (ps /\ th)))
Theorem	pm5.3 448	Theorem *5.3 of [WhiteheadRussell] p. 125.
|- (((ph /\ ps) -> ch) <-> ((ph /\ ps) -> (ph /\ ch)))
Theorem	pm5.61 449	Theorem *5.61 of [WhiteheadRussell] p. 125.
|- (((ph \/ ps) /\ -. ps) <-> (ph /\ -. ps))
Theorem	sylan 450	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ph)   =>   |- ((th /\ ps) -> ch)
Theorem	sylanb 451	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ph)   =>   |- ((th /\ ps) -> ch)
Theorem	sylanbr 452	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ph <-> th)   =>   |- ((th /\ ps) -> ch)
Theorem	sylan2 453	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ps)   =>   |- ((ph /\ th) -> ch)
Theorem	sylan2b 454	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ps)   =>   |- ((ph /\ th) -> ch)
Theorem	sylan2br 455	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ps <-> th)   =>   |- ((ph /\ th) -> ch)
Theorem	syl2an 456	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ph)   &   |- (ta -> ps)   =>   |- ((th /\ ta) -> ch)
Theorem	syl2anb 457	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ph)   &   |- (ta <-> ps)   =>   |- ((th /\ ta) -> ch)
Theorem	syl2anbr 458	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ph <-> th)   &   |- (ps <-> ta)   =>   |- ((th /\ ta) -> ch)
Theorem	syland 459	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ps))   =>   |- (ph -> ((ta /\ ch) -> th))
Theorem	sylan2d 460	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ch))   =>   |- (ph -> ((ps /\ ta) -> th))
Theorem	syl2and 461	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ps))   &   |- (ph -> (et -> ch))   =>   |- (ph -> ((ta /\ et) -> th))
Theorem	sylanl1 462	A syllogism inference.
|- (((ph /\ ps) /\ ch) -> th)   &   |- (ta -> ph)   =>   |- (((ta /\ ps) /\ ch) -> th)
Theorem	sylanl2 463	A syllogism inference.
|- (((ph /\ ps) /\ ch) -> th)   &   |- (ta -> ps)   =>   |- (((ph /\ ta) /\ ch) -> th)
Theorem	sylanr1 464	A syllogism inference.
|- ((ph /\ (ps /\ ch)) -> th)   &   |- (ta -> ps)   =>   |- ((ph /\ (ta /\ ch)) -> th)
Theorem	sylanr2 465	A syllogism inference.
|- ((ph /\ (ps /\ ch)) -> th)   &   |- (ta -> ch)   =>   |- ((ph /\ (ps /\ ta)) -> th)
Theorem	sylani 466	A syllogism inference.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ta -> ps)   =>   |- (ph -> ((ta /\ ch) -> th))
Theorem	sylan2i 467	A syllogism inference.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ta -> ch)   =>   |- (ph -> ((ps /\ ta) -> th))
Theorem	syl2ani 468	A syllogism inference.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ta -> ps)   &   |- (et -> ch)   =>   |- (ph -> ((ta /\ et) -> th))
Theorem	syldan 469	A syllogism deduction with conjoined antecents.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	sylan9 470	Nested syllogism inference conjoining dissimilar antecedents.
|- (ph -> (ps -> ch))   &   |- (th -> (ch -> ta))   =>   |- ((ph /\ th) -> (ps -> ta))
Theorem	sylan9r 471	Nested syllogism inference conjoining dissimilar antecedents.
|- (ph -> (ps -> ch))   &   |- (th -> (ch -> ta))   =>   |- ((th /\ ph) -> (ps -> ta))
Theorem	mpan9 472	Modus ponens conjoining dissimilar antecedents.
|- (ph -> ps)   &   |- (ch -> (ps -> th))   =>   |- ((ph /\ ch) -> th)
Theorem	sylanc 473	A syllogism inference combined with contraction.
|- ((ph /\ ps) -> ch)   &   |- (th -> ph)   &   |- (th -> ps)   =>   |- (th -> ch)
Theorem	sylancr 474	Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ((ph /\ ps) -> ch)   &   |- ph   &   |- (th -> ps)   =>   |- (th -> ch)
Theorem	syl2anc 475	A syllogism inference combined with contraction.
|- (((ph /\ ps) /\ (ch /\ th)) -> ta)   &   |- (et -> ph)   &   |- (et -> ps)   &   |- (et -> ch)   &   |- (et -> th)   =>   |- (et -> ta)
Theorem	sylancb 476	A syllogism inference combined with contraction.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ph)   &   |- (th <-> ps)   =>   |- (th -> ch)
Theorem	sylancbr 477	A syllogism inference combined with contraction.
|- ((ph /\ ps) -> ch)   &   |- (ph <-> th)   &   |- (ps <-> th)   =>   |- (th -> ch)
Theorem	sylancom 478	Syllogism inference with commutation of antecents.
|- ((ph /\ ps) -> ch)   &   |- ((ch /\ ps) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	pm2.61ian 479	Elimination of an antecedent.
|- ((ph /\ ps) -> ch)   &   |- ((-. ph /\ ps) -> ch)   =>   |- (ps -> ch)
Theorem	pm2.61dan 480	Elimination of an antecedent.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ -. ps) -> ch)   =>   |- (ph -> ch)
Theorem	condan 481	Proof by contradiction.
|- ((ph /\ -. ps) -> ch)   &   |- ((ph /\ -. ps) -> -. ch)   =>   |- (ph -> ps)
Theorem	abai 482	Introduce one conjunct as an antecedent to the another.
|- ((ph /\ ps) <-> (ph /\ (ph -> ps)))
Theorem	anbi2i 483	Introduce a left conjunct to both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ((ch /\ ph) <-> (ch /\ ps))
Theorem	anbi1i 484	Introduce a right conjunct to both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ((ph /\ ch) <-> (ps /\ ch))
Theorem	anbi12i 485	Conjoin both sides of two equivalences.
|- (ph <-> ps)   &   |- (ch <-> th)   =>   |- ((ph /\ ch) <-> (ps /\ th))
Theorem	pm5.53 486	Theorem *5.53 of [WhiteheadRussell] p. 125.
|- ((((ph \/ ps) \/ ch) -> th) <-> (((ph -> th) /\ (ps -> th)) /\ (ch -> th)))
Theorem	an12 487	A rearrangement of conjuncts.
|- ((ph /\ (ps /\ ch)) <-> (ps /\ (ph /\ ch)))
Theorem	an23 488	A rearrangement of conjuncts.
|- (((ph /\ ps) /\ ch) <-> ((ph /\ ch) /\ ps))
Theorem	an1s 489	Deduction rearranging conjuncts.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ps /\ (ph /\ ch)) -> th)
Theorem	ancom2s 490	Inference commuting a nested conjunction in antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ch /\ ps)) -> th)
Theorem	ancom13s 491	Deduction rearranging conjuncts.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ch /\ (ps /\ ph)) -> th)
Theorem	an1rs 492	Deduction rearranging conjuncts.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ch) /\ ps) -> th)
Theorem	ancom1s 493	Inference commuting a nested conjunction in antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ps /\ ph) /\ ch) -> th)
Theorem	ancom31s 494	Deduction rearranging conjuncts.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ch /\ ps) /\ ph) -> th)
Theorem	anabs1 495	Absorption into embedded conjunct.
|- (((ph /\ ps) /\ ph) <-> (ph /\ ps))
Theorem	anabs5 496	Absorption into embedded conjunct.
|- ((ph /\ (ph /\ ps)) <-> (ph /\ ps))
Theorem	anabs7 497	Absorption into embedded conjunct.
|- ((ps /\ (ph /\ ps)) <-> (ph /\ ps))
Theorem	anabsi5 498	Absorption of antecedent into conjunction.
|- (ph -> ((ph /\ ps) -> ch))   =>   |- ((ph /\ ps) -> ch)
Theorem	anabsi6 499	Absorption of antecedent into conjunction.
|- (ph -> ((ps /\ ph) -> ch))   =>   |- ((ph /\ ps) -> ch)
Theorem	anabsi7 500	Absorption of antecedent into conjunction.
|- (ps -> ((ph /\ ps) -> ch))   =>   |- ((ph /\ ps) -> ch)