mmtheorems7 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 601-700 - Page 7 of 175

Type	Label	Description
Statement
Theorem	3imtr3d 601	More general version of 3imtr3i 235. Useful for converting conditional definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ph -> (ps <-> th))   &   |- (ph -> (ch <-> ta))   =>   |- (ph -> (th -> ta))
Theorem	3imtr4d 602	More general version of 3imtr4i 236. Useful for converting conditional definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ph -> (th <-> ps))   &   |- (ph -> (ta <-> ch))   =>   |- (ph -> (th -> ta))
Theorem	3bitrd 603	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ps <-> ta))
Theorem	3bitrrd 604	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ta <-> ps))
Theorem	3bitr2d 605	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ps <-> ta))
Theorem	3bitr2rd 606	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> (ta <-> ps))
Theorem	3bitr3d 607	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 608	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ps <-> th))   &   |- (ph -> (ch <-> ta))   =>   |- (ph -> (ta <-> th))
Theorem	3bitr4d 609	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 610	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ps))   &   |- (ph -> (ta <-> ch))   =>   |- (ph -> (ta <-> th))
Theorem	3imtr3g 611	More general version of 3imtr3i 235. Useful for converting definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ps <-> th)   &   |- (ch <-> ta)   =>   |- (ph -> (th -> ta))
Theorem	3imtr4g 612	More general version of 3imtr4i 236. Useful for converting definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (th <-> ps)   &   |- (ta <-> ch)   =>   |- (ph -> (th -> ta))
Theorem	3bitr3g 613	More general version of 3bitr3i 198. Useful for converting definitions in a formula.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   &   |- (ch <-> ta)   =>   |- (ph -> (th <-> ta))
Theorem	3bitr4g 614	More general version of 3bitr4i 200. Useful for converting definitions in a formula.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   &   |- (ta <-> ch)   =>   |- (ph -> (th <-> ta))
Theorem	prth 615	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	pm3.48 616	Theorem *3.48 of [WhiteheadRussell] p. 114.
|- (((ph -> ps) /\ (ch -> th)) -> ((ph \/ ch) -> (ps \/ th)))
Theorem	anim12d 617	Conjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta))   =>   |- (ph -> ((ps /\ th) -> (ch /\ ta)))
Theorem	anim12ii 618	Conjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   &   |- (th -> (ps -> ta))   =>   |- ((ph /\ th) -> (ps -> (ch /\ ta)))
Theorem	anim1d 619	Add a conjunct to right of antecedent and consequent in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((ps /\ th) -> (ch /\ th)))
Theorem	anim2d 620	Add a conjunct to left of antecedent and consequent in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((th /\ ps) -> (th /\ ch)))
Theorem	pm3.45 621	Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113.
|- ((ph -> ps) -> ((ph /\ ch) -> (ps /\ ch)))
Theorem	im2anan9 622	Deduction joining nested implications to form implication of conjunctions.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> et))   =>   |- ((ph /\ th) -> ((ps /\ ta) -> (ch /\ et)))
Theorem	im2anan9r 623	Deduction joining nested implications to form implication of conjunctions.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> et))   =>   |- ((th /\ ph) -> ((ps /\ ta) -> (ch /\ et)))
Theorem	orim12d 624	Disjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta))   =>   |- (ph -> ((ps \/ th) -> (ch \/ ta)))
Theorem	orim1d 625	Disjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((ps \/ th) -> (ch \/ th)))
Theorem	orim2d 626	Disjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((th \/ ps) -> (th \/ ch)))
Theorem	orim2 627	Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97.
|- ((ps -> ch) -> ((ph \/ ps) -> (ph \/ ch)))
Theorem	pm2.38 628	Theorem *2.38 of [WhiteheadRussell] p. 105.
|- ((ps -> ch) -> ((ps \/ ph) -> (ch \/ ph)))
Theorem	pm2.36 629	Theorem *2.36 of [WhiteheadRussell] p. 105.
|- ((ps -> ch) -> ((ph \/ ps) -> (ch \/ ph)))
Theorem	pm2.37 630	Theorem *2.37 of [WhiteheadRussell] p. 105.
|- ((ps -> ch) -> ((ps \/ ph) -> (ph \/ ch)))
Theorem	pm2.73 631	Theorem *2.73 of [WhiteheadRussell] p. 108.
|- ((ph -> ps) -> (((ph \/ ps) \/ ch) -> (ps \/ ch)))
Theorem	pm2.74 632	Theorem *2.74 of [WhiteheadRussell] p. 108. (The proof was shortened by Andrew Salmon, 7-May-2011.)
|- ((ps -> ph) -> (((ph \/ ps) \/ ch) -> (ph \/ ch)))
Theorem	pm2.74OLD 633	Theorem *2.74 of [WhiteheadRussell] p. 108.
|- ((ps -> ph) -> (((ph \/ ps) \/ ch) -> (ph \/ ch)))
Theorem	pm2.75 634	Theorem *2.75 of [WhiteheadRussell] p. 108.
|- ((ph \/ ps) -> ((ph \/ (ps -> ch)) -> (ph \/ ch)))
Theorem	pm2.76 635	Theorem *2.76 of [WhiteheadRussell] p. 108.
|- ((ph \/ (ps -> ch)) -> ((ph \/ ps) -> (ph \/ ch)))
Theorem	pm2.8 636	Theorem *2.8 of [WhiteheadRussell] p. 108.
|- ((ps \/ ch) -> ((-. ch \/ th) -> (ps \/ th)))
Theorem	pm2.81 637	Theorem *2.81 of [WhiteheadRussell] p. 108.
|- ((ps -> (ch -> th)) -> ((ph \/ ps) -> ((ph \/ ch) -> (ph \/ th))))
Theorem	pm2.82 638	Theorem *2.82 of [WhiteheadRussell] p. 108.
|- (((ph \/ ps) \/ ch) -> (((ph \/ -. ch) \/ th) -> ((ph \/ ps) \/ th)))
Theorem	pm2.85 639	Theorem *2.85 of [WhiteheadRussell] p. 108.
|- (((ph \/ ps) -> (ph \/ ch)) -> (ph \/ (ps -> ch)))
Theorem	pm3.2ni 640	Infer negated disjunction of negated premises.
|- -. ph   &   |- -. ps   =>   |- -. (ph \/ ps)
Theorem	orabs 641	Absorption of redundant internal disjunct. Compare Theorem *4.45 of [WhiteheadRussell] p. 119.
|- (ph <-> ((ph \/ ps) /\ ph))
Theorem	oranabs 642	Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton 23-Jun-2005.)
|- (((ph \/ -. ps) /\ ps) <-> (ph /\ ps))
Theorem	pm5.74 643	Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126.
|- ((ph -> (ps <-> ch)) <-> ((ph -> ps) <-> (ph -> ch)))
Theorem	pm5.74i 644	Distribution of implication over biconditional (inference rule).
|- (ph -> (ps <-> ch))   =>   |- ((ph -> ps) <-> (ph -> ch))
Theorem	pm5.74d 645	Distribution of implication over biconditional (deduction rule).
|- (ph -> (ps -> (ch <-> th)))   =>   |- (ph -> ((ps -> ch) <-> (ps -> th)))
Theorem	pm5.74da 646	Distribution of implication over biconditional (deduction rule).
|- ((ph /\ ps) -> (ch <-> th))   =>   |- (ph -> ((ps -> ch) <-> (ps -> th)))
Theorem	pm5.74ri 647	Distribution of implication over biconditional (reverse inference rule).
|- ((ph -> ps) <-> (ph -> ch))   =>   |- (ph -> (ps <-> ch))
Theorem	pm5.74rd 648	Distribution of implication over biconditional (deduction rule).
|- (ph -> ((ps -> ch) <-> (ps -> th)))   =>   |- (ph -> (ps -> (ch <-> th)))
Theorem	mpbidi 649	A deduction from a biconditional, related to modus ponens.
|- (th -> (ph -> ps))   &   |- (ph -> (ps <-> ch))   =>   |- (th -> (ph -> ch))
Theorem	ibib 650	Implication in terms of implication and biconditional.
|- ((ph -> ps) <-> (ph -> (ph <-> ps)))
Theorem	ibibr 651	Implication in terms of implication and biconditional.
|- ((ph -> ps) <-> (ph -> (ps <-> ph)))
Theorem	ibi 652	Inference that converts a biconditional implied by one of its arguments, into an implication.
|- (ph -> (ph <-> ps))   =>   |- (ph -> ps)
Theorem	ibir 653	Inference that converts a biconditional implied by one of its arguments, into an implication.
|- (ph -> (ps <-> ph))   =>   |- (ph -> ps)
Theorem	ibd 654	Deduction that converts a biconditional implied by one of its arguments, into an implication.
|- (ph -> (ps -> (ps <-> ch)))   =>   |- (ph -> (ps -> ch))
Theorem	pm5.501 655	Theorem *5.501 of [WhiteheadRussell] p. 125.
|- (ph -> (ps <-> (ph <-> ps)))
Theorem	ordi 656	Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (The proof was shortened by Andrew Salmon, 7-May-2011.)
|- ((ph \/ (ps /\ ch)) <-> ((ph \/ ps) /\ (ph \/ ch)))
Theorem	ordiOLD 657	Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119.
|- ((ph \/ (ps /\ ch)) <-> ((ph \/ ps) /\ (ph \/ ch)))
Theorem	ordir 658	Distributive law for disjunction.
|- (((ph /\ ps) \/ ch) <-> ((ph \/ ch) /\ (ps \/ ch)))
Theorem	jcab 659	Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121.
|- ((ph -> (ps /\ ch)) <-> ((ph -> ps) /\ (ph -> ch)))
Theorem	pm4.76 660	Theorem *4.76 of [WhiteheadRussell] p. 121.
|- (((ph -> ps) /\ (ph -> ch)) <-> (ph -> (ps /\ ch)))
Theorem	jcad 661	Deduction conjoining the consequents of two implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (ps -> th))   =>   |- (ph -> (ps -> (ch /\ th)))
Theorem	jctild 662	Deduction conjoining a theorem to left of consequent in an implication.
|- (ph -> (ps -> ch))   &   |- (ph -> th)   =>   |- (ph -> (ps -> (th /\ ch)))
Theorem	jctird 663	Deduction conjoining a theorem to right of consequent in an implication.
|- (ph -> (ps -> ch))   &   |- (ph -> th)   =>   |- (ph -> (ps -> (ch /\ th)))
Theorem	pm3.43 664	Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113.
|- (((ph -> ps) /\ (ph -> ch)) -> (ph -> (ps /\ ch)))
Theorem	andi 665	Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118.
|- ((ph /\ (ps \/ ch)) <-> ((ph /\ ps) \/ (ph /\ ch)))
Theorem	andir 666	Distributive law for conjunction.
|- (((ph \/ ps) /\ ch) <-> ((ph /\ ch) \/ (ps /\ ch)))
Theorem	orddi 667	Double distributive law for disjunction.
|- (((ph /\ ps) \/ (ch /\ th)) <-> (((ph \/ ch) /\ (ph \/ th)) /\ ((ps \/ ch) /\ (ps \/ th))))
Theorem	anddi 668	Double distributive law for conjunction.
|- (((ph \/ ps) /\ (ch \/ th)) <-> (((ph /\ ch) \/ (ph /\ th)) \/ ((ps /\ ch) \/ (ps /\ th))))
Theorem	bibi2i 669	Inference adding a biconditional to the left in an equivalence. (The proof was shortened by Andrew Salmon, 7-May-2011.)
|- (ph <-> ps)   =>   |- ((ch <-> ph) <-> (ch <-> ps))
Theorem	bibi2iOLD 670	Inference adding a biconditional to the left in an equivalence.
|- (ph <-> ps)   =>   |- ((ch <-> ph) <-> (ch <-> ps))
Theorem	bibi1i 671	Inference adding a biconditional to the right in an equivalence.
|- (ph <-> ps)   =>   |- ((ph <-> ch) <-> (ps <-> ch))
Theorem	bibi12i 672	The equivalence of two equivalences.
|- (ph <-> ps)   &   |- (ch <-> th)   =>   |- ((ph <-> ch) <-> (ps <-> th))
Theorem	notbid 673	Deduction negating both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> (-. ps <-> -. ch))
Theorem	imbi2d 674	Deduction adding an antecedent to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th -> ps) <-> (th -> ch)))
Theorem	imbi1d 675	Deduction adding a consequent to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps -> th) <-> (ch -> th)))
Theorem	orbi2d 676	Deduction adding a left disjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th \/ ps) <-> (th \/ ch)))
Theorem	orbi1d 677	Deduction adding a right disjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps \/ th) <-> (ch \/ th)))
Theorem	anbi2d 678	Deduction adding a left conjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th /\ ps) <-> (th /\ ch)))
Theorem	anbi1d 679	Deduction adding a right conjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps /\ th) <-> (ch /\ th)))
Theorem	bibi2d 680	Deduction adding a biconditional to the left in an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th <-> ps) <-> (th <-> ch)))
Theorem	bibi1d 681	Deduction adding a biconditional to the right in an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps <-> th) <-> (ch <-> th)))
Theorem	orbi1 682	Theorem *4.37 of [WhiteheadRussell] p. 118.
|- ((ph <-> ps) -> ((ph \/ ch) <-> (ps \/ ch)))
Theorem	anbi1 683	Theorem *4.36 of [WhiteheadRussell] p. 118.
|- ((ph <-> ps) -> ((ph /\ ch) <-> (ps /\ ch)))
Theorem	bitr 684	Theorem *4.22 of [WhiteheadRussell] p. 117.
|- (((ph <-> ps) /\ (ps <-> ch)) -> (ph <-> ch))
Theorem	imbi1 685	Theorem *4.84 of [WhiteheadRussell] p. 122.
|- ((ph <-> ps) -> ((ph -> ch) <-> (ps -> ch)))
Theorem	imbi2 686	Theorem *4.85 of [WhiteheadRussell] p. 122.
|- ((ph <-> ps) -> ((ch -> ph) <-> (ch -> ps)))
Theorem	bibi1 687	Theorem *4.86 of [WhiteheadRussell] p. 122.
|- ((ph <-> ps) -> ((ph <-> ch) <-> (ps <-> ch)))
Theorem	imbi12d 688	Deduction joining two equivalences to form equivalence of implications.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps -> th) <-> (ch -> ta)))
Theorem	orbi12d 689	Deduction joining two equivalences to form equivalence of disjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps \/ th) <-> (ch \/ ta)))
Theorem	anbi12d 690	Deduction joining two equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps /\ th) <-> (ch /\ ta)))
Theorem	bibi12d 691	Deduction joining two equivalences to form equivalence of biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps <-> th) <-> (ch <-> ta)))
Theorem	pm4.39 692	Theorem *4.39 of [WhiteheadRussell] p. 118.
|- (((ph <-> ch) /\ (ps <-> th)) -> ((ph \/ ps) <-> (ch \/ th)))
Theorem	pm4.38 693	Theorem *4.38 of [WhiteheadRussell] p. 118.
|- (((ph <-> ch) /\ (ps <-> th)) -> ((ph /\ ps) <-> (ch /\ th)))
Theorem	bi2anan9 694	Deduction joining two equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (th -> (ta <-> et))   =>   |- ((ph /\ th) -> ((ps /\ ta) <-> (ch /\ et)))
Theorem	bi2anan9r 695	Deduction joining two equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (th -> (ta <-> et))   =>   |- ((th /\ ph) -> ((ps /\ ta) <-> (ch /\ et)))
Theorem	bi2bian9 696	Deduction joining two biconditionals with different antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ta <-> et))   =>   |- ((ph /\ th) -> ((ps <-> ta) <-> (ch <-> et)))
Theorem	pm4.71 697	Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120.
|- ((ph -> ps) <-> (ph <-> (ph /\ ps)))
Theorem	pm4.71r 698	Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed).
|- ((ph -> ps) <-> (ph <-> (ps /\ ph)))
Theorem	pm4.71i 699	Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120.
|- (ph -> ps)   =>   |- (ph <-> (ph /\ ps))
Theorem	pm4.71ri 700	Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed).
|- (ph -> ps)   =>   |- (ph <-> (ps /\ ph))