mmtheorems6 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 501-600 - Page 6 of 108

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	dfbi2 514	A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49.
|- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
Theorem	dfbi 515	Definition df-bi 147 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 516	Deduce an equivalence from two implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (ch -> ps))   =>   |- (ph -> (ps <-> ch))
Theorem	impbid1 517	Infer an equivalence from two implications.
|- (ph -> (ps -> ch))   &   |- (ch -> ps)   =>   |- (ph -> (ps <-> ch))
Theorem	impbid2 518	Infer an equivalence from two implications.
|- (ps -> ch)   &   |- (ph -> (ch -> ps))   =>   |- (ph -> (ps <-> ch))
Theorem	impbida 519	Deduce an equivalence from two implications.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ ch) -> ps)   =>   |- (ph -> (ps <-> ch))
Theorem	bicom 520	Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117.
|- ((ph <-> ps) <-> (ps <-> ph))
Theorem	bicomd 521	Commute two sides of a biconditional in a deduction.
|- (ph -> (ps <-> ch))   =>   |- (ph -> (ch <-> ps))
Theorem	pm4.11 522	Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117.
|- ((ph <-> ps) <-> (-. ph <-> -. ps))
Theorem	con4bii 523	A contraposition inference.
|- (-. ph <-> -. ps)   =>   |- (ph <-> ps)
Theorem	con4bid 524	A contraposition deduction.
|- (ph -> (-. ps <-> -. ch))   =>   |- (ph -> (ps <-> ch))
Theorem	con2bi 525	Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117.
|- ((ph <-> -. ps) <-> (ps <-> -. ph))
Theorem	con2bid 526	A contraposition deduction.
|- (ph -> (ps <-> -. ch))   =>   |- (ph -> (ch <-> -. ps))
Theorem	con1bid 527	A contraposition deduction.
|- (ph -> (-. ps <-> ch))   =>   |- (ph -> (-. ch <-> ps))
Theorem	bitrd 528	Deduction form of bitr 173.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   =>   |- (ph -> (ps <-> th))
Theorem	bitr2d 529	Deduction form of bitr2 174.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   =>   |- (ph -> (th <-> ps))
Theorem	bitr3d 530	Deduction form of bitr3 175.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ps <-> th))   =>   |- (ph -> (ch <-> th))
Theorem	bitr4d 531	Deduction form of bitr4 176.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   =>   |- (ph -> (ps <-> th))
Theorem	syl5bb 532	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   =>   |- (ph -> (th <-> ch))
Theorem	syl5rbb 533	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   =>   |- (ph -> (ch <-> th))
Theorem	syl5bbr 534	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   =>   |- (ph -> (th <-> ch))
Theorem	syl5rbbr 535	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   =>   |- (ph -> (ch <-> th))
Theorem	syl6bb 536	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ch <-> th)   =>   |- (ph -> (ps <-> th))
Theorem	syl6rbb 537	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ch <-> th)   =>   |- (ph -> (th <-> ps))
Theorem	syl6bbr 538	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ch)   =>   |- (ph -> (ps <-> th))
Theorem	syl6rbbr 539	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ch)   =>   |- (ph -> (th <-> ps))
Theorem	sylan9bb 540	Nested syllogism inference conjoining dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta))   =>   |- ((ph /\ th) -> (ps <-> ta))
Theorem	sylan9bbr 541	Nested syllogism inference conjoining dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta))   =>   |- ((th /\ ph) -> (ps <-> ta))
Theorem	3imtr3d 542	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 543	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 544	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ps <-> ta))
Theorem	3bitrrd 545	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ta <-> ps))
Theorem	3bitr2d 546	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ps <-> ta))
Theorem	3bitr2rd 547	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ta <-> ps))
Theorem	3bitr3d 548	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 549	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ps <-> th))   &   |- (ph -> (ch <-> ta))   =>   |- (ph -> (ta <-> th))
Theorem	3bitr4d 550	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 551	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th