mmtheorems6 - 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 - 501-600 - Page 6 of 123

Type	Label	Description
Statement
Theorem	anabsi8 501	Absorption of antecedent into conjunction.
|- (ps -> ((ps /\ ph) -> ch))   =>   |- ((ph /\ ps) -> ch)
Theorem	anabss1 502	Absorption of antecedent into conjunction.
|- (((ph /\ ps) /\ ph) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	anabss3 503	Absorption of antecedent into conjunction.
|- (((ph /\ ps) /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	anabss4 504	Absorption of antecedent into conjunction.
|- (((ps /\ ph) /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	anabss5 505	Absorption of antecedent into conjunction.
|- ((ph /\ (ph /\ ps)) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	anabss7 506	Absorption of antecedent into conjunction.
|- ((ps /\ (ph /\ ps)) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	anabsan 507	Absorption of antecedent with conjunction.
|- (((ph /\ ph) /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	anabsan2 508	Absorption of antecedent with conjunction.
|- ((ph /\ (ps /\ ps)) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	an4 509	Rearrangement of 4 conjuncts.
|- (((ph /\ ps) /\ (ch /\ th)) <-> ((ph /\ ch) /\ (ps /\ th)))
Theorem	an42 510	Rearrangement of 4 conjuncts.
|- (((ph /\ ps) /\ (ch /\ th)) <-> ((ph /\ ch) /\ (th /\ ps)))
Theorem	an4s 511	Inference rearranging 4 conjuncts in antecedent.
|- (((ph /\ ps) /\ (ch /\ th)) -> ta)   =>   |- (((ph /\ ch) /\ (ps /\ th)) -> ta)
Theorem	an42s 512	Inference rearranging 4 conjuncts in antecedent.
|- (((ph /\ ps) /\ (ch /\ th)) -> ta)   =>   |- (((ph /\ ch) /\ (th /\ ps)) -> ta)
Theorem	anandi 513	Distribution of conjunction over conjunction.
|- ((ph /\ (ps /\ ch)) <-> ((ph /\ ps) /\ (ph /\ ch)))
Theorem	anandir 514	Distribution of conjunction over conjunction.
|- (((ph /\ ps) /\ ch) <-> ((ph /\ ch) /\ (ps /\ ch)))
Theorem	anandis 515	Inference that undistributes conjunction in the antecedent.
|- (((ph /\ ps) /\ (ph /\ ch)) -> ta)   =>   |- ((ph /\ (ps /\ ch)) -> ta)
Theorem	anandirs 516	Inference that undistributes conjunction in the antecedent.
|- (((ph /\ ch) /\ (ps /\ ch)) -> ta)   =>   |- (((ph /\ ps) /\ ch) -> ta)
Theorem	dfbi2 517	A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49.
|- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
Theorem	dfbi 518	Definition df-bi 145 rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional.
|- (((ph <-> ps) -> ((ph -> ps) /\ (ps -> ph))) /\ (((ph -> ps) /\ (ps -> ph)) -> (ph <-> ps)))
Theorem	impbid 519	Deduce an equivalence from two implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (ch -> ps))   =>   |- (ph -> (ps <-> ch))
Theorem	impbid1 520	Infer an equivalence from two implications.
|- (ph -> (ps -> ch))   &   |- (ch -> ps)   =>   |- (ph -> (ps <-> ch))
Theorem	impbid2 521	Infer an equivalence from two implications.
|- (ps -> ch)   &   |- (ph -> (ch -> ps))   =>   |- (ph -> (ps <-> ch))
Theorem	impbida 522	Deduce an equivalence from two implications.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ ch) -> ps)   =>   |- (ph -> (ps <-> ch))
Theorem	bicom 523	Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117.
|- ((ph <-> ps) <-> (ps <-> ph))
Theorem	bicomd 524	Commute two sides of a biconditional in a deduction.
|- (ph -> (ps <-> ch))   =>   |- (ph -> (ch <-> ps))
Theorem	notbi 525	Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117.
|- ((ph <-> ps) <-> (-. ph <-> -. ps))
Theorem	con4bii 526	A contraposition inference.
|- (-. ph <-> -. ps)   =>   |- (ph <-> ps)
Theorem	con4bid 527	A contraposition deduction.
|- (ph -> (-. ps <-> -. ch))   =>   |- (ph -> (ps <-> ch))
Theorem	con2bi 528	Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117.
|- ((ph <-> -. ps) <-> (ps <-> -. ph))
Theorem	con2bid 529	A contraposition deduction.
|- (ph -> (ps <-> -. ch))   =>   |- (ph -> (ch <-> -. ps))
Theorem	con1bid 530	A contraposition deduction.
|- (ph -> (-. ps <-> ch))   =>   |- (ph -> (-. ch <-> ps))
Theorem	bitrd 531	Deduction form of bitri 171.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   =>   |- (ph -> (ps <-> th))
Theorem	bitr2d 532	Deduction form of bitr2i 172.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   =>   |- (ph -> (th <-> ps))
Theorem	bitr3d 533	Deduction form of bitr3i 173.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ps <-> th))   =>   |- (ph -> (ch <-> th))
Theorem	bitr4d 534	Deduction form of bitr4i 174.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   =>   |- (ph -> (ps <-> th))
Theorem	syl5bb 535	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   =>   |- (ph -> (th <-> ch))
Theorem	syl5rbb 536	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   =>   |- (ph -> (ch <-> th))
Theorem	syl5bbr 537	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   =>   |- (ph -> (th <-> ch))
Theorem	syl5rbbr 538	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   =>   |- (ph -> (ch <-> th))
Theorem	syl6bb 539	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ch <-> th)   =>   |- (ph -> (ps <-> th))
Theorem	syl6rbb 540	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ch <-> th)   =>   |- (ph -> (th <-> ps))
Theorem	syl6bbr 541	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ch)   =>   |- (ph -> (ps <-> th))
Theorem	syl6rbbr 542	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ch)   =>   |- (ph -> (th <-> ps))
Theorem	sylan9bb 543	Nested syllogism inference conjoining dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta))   =>   |- ((ph /\ th) -> (ps <-> ta))
Theorem	sylan9bbr 544	Nested syllogism inference conjoining dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta))   =>   |- ((th /\ ph) -> (ps <-> ta))
Theorem	3imtr3d 545	More general version of 3imtr3i 216. Useful for converting conditional definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ph -> (ps <-> th))   &   |- (ph -> (ch <-> ta))   =>   |- (ph -> (th -> ta))
Theorem	3imtr4d 546	More general version of 3imtr4i 217. Useful for converting conditional definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ph -> (th <-> ps))   &   |- (ph -> (ta <-> ch))   =>   |- (ph -> (th -> ta))
Theorem	3bitrd 547	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ps <-> ta))
Theorem	3bitrrd 548	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ta <-> ps))
Theorem	3bitr2d 549	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ps <-> ta))
Theorem	3bitr2rd 550	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ta <-> ps))
Theorem	3bitr3d 551	Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ps <-> th))   &   |- (ph -> (ch <-> ta))   =>   |- (ph -> (th <-> ta))
Theorem	3bitr3rd 552	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ps <-> th))   &   |- (ph -> (ch <-> ta))   =>   |- (ph -> (ta <-> th))
Theorem	3bitr4d 553	Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ps))   &   |- (ph -> (ta <-> ch))   =>   |- (ph -> (th <-> ta))
Theorem	3bitr4rd 554	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ps))   &   |- (ph -> (ta <-> ch))   =>   |- (ph -> (ta <-> th))
Theorem	3imtr3g 555	More general version of 3imtr3i 216. Useful for converting definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ps <-> th)   &   |- (ch <-> ta)   =>   |- (ph -> (th -> ta))
Theorem	3imtr4g 556	More general version of 3imtr4i 217. Useful for converting definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (th <-> ps)   &   |- (ta <-> ch)   =>   |- (ph -> (th -> ta))
Theorem	3bitr3g 557	More general version of 3bitr3i 179. Useful for converting definitions in a formula.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   &   |- (ch <-> ta)   =>   |- (ph -> (th <-> ta))
Theorem	3bitr4g 558	More general version of 3bitr4i 181. Useful for converting definitions in a formula.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   &   |- (ta <-> ch)   =>   |- (ph -> (th <-> ta))
Theorem	prth 559	Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it 'praeclarum theorema' (splendid theorem).
|- (((ph -> ps) /\ (ch -> th)) -> ((ph /\ ch) -> (ps /\ th)))
Theorem	pm3.48 560	Theorem *3.48 of [WhiteheadRussell] p. 114.
|- (((ph -> ps) /\ (ch -> th)) -> ((ph \/ ch) -> (ps \/ th)))
Theorem	anim12d 561	Conjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta))   =>   |- (ph -> ((ps /\ th) -> (ch /\ ta)))
Theorem	anim12ii 562	Conjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   &   |- (th -> (ps -> ta))   =>   |- ((ph /\ th) -> (ps -> (ch /\ ta)))
Theorem	anim1d 563	Add a conjunct to right of antecedent and consequent in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((ps /\ th) -> (ch /\ th)))
Theorem	anim2d 564	Add a conjunct to left of antecedent and consequent in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((th /\ ps) -> (th /\ ch)))
Theorem	pm3.45 565	Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113.
|- ((ph -> ps) -> ((ph /\ ch) -> (ps /\ ch)))
Theorem	im2anan9 566	Deduction joining nested implications to form implication of conjunctions.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> et))   =>   |- ((ph /\ th) -> ((ps /\ ta) -> (ch /\ et)))
Theorem	im2anan9r 567	Deduction joining nested implications to form implication of conjunctions.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> et))   =>   |- ((th /\ ph) -> ((ps /\ ta) -> (ch /\ et)))
Theorem	orim12d 568	Disjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta))   =>   |- (ph -> ((ps \/ th) -> (ch \/ ta)))
Theorem	orim1d 569	Disjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((ps \/ th) -> (ch \/ th)))
Theorem	orim2d 570	Disjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((th \/ ps) -> (th \/ ch)))
Theorem	orim2 571	Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97.
|- ((ps -> ch) -> ((ph \/ ps) -> (ph \/ ch)))
Theorem	pm2.38 572	Theorem *2.38 of [WhiteheadRussell] p. 105.
|- ((ps -> ch) -> ((ps \/ ph) -> (ch \/ ph)))
Theorem	pm2.36 573	Theorem *2.36 of [WhiteheadRussell] p. 105.
|- ((ps -> ch) -> ((ph \/ ps) -> (ch \/ ph)))
Theorem	pm2.37 574	Theorem *2.37 of [WhiteheadRussell] p. 105.
|- ((ps -> ch) -> ((ps \/ ph) -> (ph \/ ch)))
Theorem	pm2.73 575	Theorem *2.73 of [WhiteheadRussell] p. 108.
|- ((ph -> ps) -> (((ph \/ ps) \/ ch) -> (ps \/ ch)))
Theorem	pm2.74 576	Theorem *2.74 of [WhiteheadRussell] p. 108.
|- ((ps -> ph) -> (((ph \/ ps) \/ ch) -> (ph \/ ch)))
Theorem	pm2.75 577	Theorem *2.75 of [WhiteheadRussell] p. 108.
|- ((ph \/ ps) -> ((ph \/ (ps -> ch)) -> (ph \/ ch)))
Theorem	pm2.76 578	Theorem *2.76 of [WhiteheadRussell] p. 108.
|- ((ph \/ (ps -> ch)) -> ((ph \/ ps) -> (ph \/ ch)))
Theorem	pm2.8 579	Theorem *2.8 of [WhiteheadRussell] p. 108.
|- ((ps \/ ch) -> ((-. ch \/ th) -> (ps \/ th)))
Theorem	pm2.81 580	Theorem *2.81 of [WhiteheadRussell] p. 108.
|- ((ps -> (ch -> th)) -> ((ph \/ ps) -> ((ph \/ ch) -> (ph \/ th))))
Theorem	pm2.82 581	Theorem *2.82 of [WhiteheadRussell] p. 108.
|- (((ph \/ ps) \/ ch) -> (((ph \/ -. ch) \/ th) -> ((ph \/ ps) \/ th)))
Theorem	pm2.85 582	Theorem *2.85 of [WhiteheadRussell] p. 108.
|- (((ph \/ ps) -> (ph \/ ch)) -> (ph \/ (ps -> ch)))
Theorem	pm3.2ni 583	Infer negated disjunction of negated premises.
|- -. ph   &   |- -. ps   =>   |- -. (ph \/ ps)
Theorem	orabs 584	Absorption of redundant internal disjunct. Compare Theorem *4.45 of [WhiteheadRussell] p. 119.
|- (ph <-> ((ph \/ ps) /\ ph))
Theorem	oranabs 585	Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton 23-Jun-2005.)
|- (((ph \/ -. ps) /\ ps) <-> (ph /\ ps))
Theorem	pm5.74 586	Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126.
|- ((ph -> (ps <-> ch)) <-> ((ph -> ps) <-> (ph -> ch)))
Theorem	pm5.74i 587	Distribution of implication over biconditional (inference rule).
|- (ph -> (ps <-> ch))   =>   |- ((ph -> ps) <-> (ph -> ch))
Theorem	pm5.74d 588	Distribution of implication over biconditional (deduction rule).
|- (ph -> (ps -> (ch <-> th)))   =>   |- (ph -> ((ps -> ch) <-> (ps -> th)))
Theorem	pm5.74da 589	Distribution of implication over biconditional (deduction rule).
|- ((ph /\ ps) -> (ch <-> th))   =>   |- (ph -> ((ps -> ch) <-> (ps -> th)))
Theorem	pm5.74ri 590	Distribution of implication over biconditional (reverse inference rule).
|- ((ph -> ps) <-> (ph -> ch))   =>   |- (ph -> (ps <-> ch))
Theorem	pm5.74rd 591	Distribution of implication over biconditional (deduction rule).
|- (ph -> ((ps -> ch) <-> (ps -> th)))   =>   |- (ph -> (ps -> (ch <-> th)))
Theorem	mpbidi 592	A deduction from a biconditional, related to modus ponens.
|- (th -> (ph -> ps))   &   |- (ph -> (ps <-> ch))   =>   |- (th -> (ph -> ch))
Theorem	ibib 593	Implication in terms of implication and biconditional.
|- ((ph -> ps) <-> (ph -> (ph <-> ps)))
Theorem	ibibr 594	Implication in terms of implication and biconditional.
|- ((ph -> ps) <-> (ph -> (ps <-> ph)))
Theorem	ibi 595	Inference that converts a biconditional implied by one of its arguments, into an implication.
|- (ph -> (ph <-> ps))   =>   |- (ph -> ps)
Theorem	ibir 596	Inference that converts a biconditional implied by one of its arguments, into an implication.
|- (ph -> (ps <-> ph))   =>   |- (ph -> ps)
Theorem	ibd 597	Deduction that converts a biconditional implied by one of its arguments, into an implication.
|- (ph -> (ps -> (ps <-> ch)))   =>   |- (ph -> (ps -> ch))
Theorem	pm5.501 598	Theorem *5.501 of [WhiteheadRussell] p. 125.
|- (ph -> (ps <-> (ph <-> ps)))
Theorem	ordi 599	Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119.
|- ((ph \/ (ps /\ ch)) <-> ((ph \/ ps) /\ (ph \/ ch)))
Theorem	ordir 600	Distributive law for disjunction.
|- (((ph /\ ps) \/ ch) <-> ((ph \/ ch) /\ (ps \/ ch)))