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 10601-10688

Statement List for Metamath Proof Explorer - 501-600 - Page 6 of 107

Type	Label	Description
Statement
Theorem	anabss4 501	Absorption of antecedent into conjunction.
|- (((ps /\ ph) /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	anabss5 502	Absorption of antecedent into conjunction.
|- ((ph /\ (ph /\ ps)) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	anabss7 503	Absorption of antecedent into conjunction.
|- ((ps /\ (ph /\ ps)) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	anabsan 504	Absorption of antecedent with conjunction.
|- (((ph /\ ph) /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	anabsan2 505	Absorption of antecedent with conjunction.
|- ((ph /\ (ps /\ ps)) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	an4 506	Rearrangement of 4 conjuncts.
|- (((ph /\ ps) /\ (ch /\ th)) <-> ((ph /\ ch) /\ (ps /\ th)))
Theorem	an42 507	Rearrangement of 4 conjuncts.
|- (((ph /\ ps) /\ (ch /\ th)) <-> ((ph /\ ch) /\ (th /\ ps)))
Theorem	an4s 508	Inference rearranging 4 conjuncts in antecedent.
|- (((ph /\ ps) /\ (ch /\ th)) -> ta)   =>   |- (((ph /\ ch) /\ (ps /\ th)) -> ta)
Theorem	an42s 509	Inference rearranging 4 conjuncts in antecedent.
|- (((ph /\ ps) /\ (ch /\ th)) -> ta)   =>   |- (((ph /\ ch) /\ (th /\ ps)) -> ta)
Theorem	anandi 510	Distribution of conjunction over conjunction.
|- ((ph /\ (ps /\ ch)) <-> ((ph /\ ps) /\ (ph /\ ch)))
Theorem	anandir 511	Distribution of conjunction over conjunction.
|- (((ph /\ ps) /\ ch) <-> ((ph /\ ch) /\ (ps /\ ch)))
Theorem	anandis 512	Inference that undistributes conjunction in the antecedent.
|- (((ph /\ ps) /\ (ph /\ ch)) -> ta)   =>   |- ((ph /\ (ps /\ ch)) -> ta)
Theorem	anandirs 513	Inference that undistributes conjunction in the antecedent.
|- (((ph /\ ch) /\ (ps /\ ch)) -> ta)   =>   |- (((ph /\ ps) /\ ch) -> ta)
Theorem	bi 514	A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49.
|- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
Theorem	impbid 515	Deduce an equivalence from two implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (ch -> ps))   =>   |- (ph -> (ps <-> ch))
Theorem	impbid1 516	Infer an equivalence from two implications.
|- (ph -> (ps -> ch))   &   |- (ch -> ps)   =>   |- (ph -> (ps <-> ch))
Theorem	impbid2 517	Infer an equivalence from two implications.
|- (ps -> ch)   &   |- (ph -> (ch -> ps))   =>   |- (ph -> (ps <-> ch))
Theorem	impbida 518	Deduce an equivalence from two implications.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ ch) -> ps)   =>   |- (ph -> (ps <-> ch))
Theorem	bicom 519	Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117.
|- ((ph <-> ps) <-> (ps <-> ph))
Theorem	bicomd 520	Commute two sides of a biconditional in a deduction.
|- (ph -> (ps <-> ch))   =>   |- (ph -> (ch <-> ps))
Theorem	pm4.11 521	Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117.
|- ((ph <-> ps) <-> (-. ph <-> -. ps))
Theorem	con4bii 522	A contraposition inference.
|- (-. ph <-> -. ps)   =>   |- (ph <-> ps)
Theorem	con4bid 523	A contraposition deduction.
|- (ph -> (-. ps <-> -. ch))   =>   |- (ph -> (ps <-> ch))
Theorem	con2bi 524	Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117.
|- ((ph <-> -. ps) <-> (ps <-> -. ph))
Theorem	con2bid 525	A contraposition deduction.
|- (ph -> (ps <-> -. ch))   =>   |- (ph -> (ch <-> -. ps))
Theorem	con1bid 526	A contraposition deduction.
|- (ph -> (-. ps <-> ch))   =>   |- (ph -> (-. ch <-> ps))
Theorem	bitrd 527	Deduction form of bitr 173.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   =>   |- (ph -> (ps <-> th))
Theorem	bitr2d 528	Deduction form of bitr2 174.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   =>   |- (ph -> (th <-> ps))
Theorem	bitr3d 529	Deduction form of bitr3 175.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ps <-> th))   =>   |- (ph -> (ch <-> th))
Theorem	bitr4d 530	Deduction form of bitr4 176.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   =>   |- (ph -> (ps <-> th))
Theorem	syl5bb 531	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   =>   |- (ph -> (th <-> ch))
Theorem	syl5rbb 532	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   =>   |- (ph -> (ch <-> th))
Theorem	syl5bbr 533	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   =>   |- (ph -> (th <-> ch))
Theorem	syl5rbbr 534	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   =>   |- (ph -> (ch <-> th))
Theorem	syl6bb 535	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ch <-> th)   =>   |- (ph -> (ps <-> th))
Theorem	syl6rbb 536	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ch <-> th)   =>   |- (ph -> (th <-> ps))
Theorem	syl6bbr 537	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ch)   =>   |- (ph -> (ps <-> th))
Theorem	syl6rbbr 538	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ch)   =>   |- (ph -> (th <-> ps))
Theorem	sylan9bb 539	Nested syllogism inference conjoining dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta))   =>   |- ((ph /\ th) -> (ps <-> ta))
Theorem	sylan9bbr 540	Nested syllogism inference conjoining dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta))   =>   |- ((th /\ ph) -> (ps <-> ta))
Theorem	3imtr3d 541	More general version of 3imtr3 218. Useful for converting conditional definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ph -> (ps <-> th))   &   |- (ph -> (ch <-> ta))   =>   |- (ph -> (th -> ta))
Theorem	3imtr4d 542	More general version of 3imtr4 219. Useful for converting conditional definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ph -> (th <-> ps))   &   |- (ph -> (ta <-> ch))   =>   |- (ph -> (th -> ta))
Theorem	3bitrd 543	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ps <-> ta))
Theorem	3bitrrd 544	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ta <-> ps))
Theorem	3bitr2d 545	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ps <-> ta))
Theorem	3bitr2rd 546	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ta <-> ps))
Theorem	3bitr3d 547	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 548	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ps <-> th))   &   |- (ph -> (ch <-> ta))   =>   |- (ph -> (ta <-> th))
Theorem	3bitr4d 549	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 550	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ps))   &   |- (ph -> (ta <-> ch))   =>   |- (ph -> (ta <-> th))
Theorem	3imtr3g 551	More general version of 3imtr3 218. Useful for converting definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ps <-> th)   &   |- (ch <-> ta) &nb