mmtheorems7 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 601-700 - Page 7 of 123

Type	Label	Description
Statement
Theorem	jcab 601	Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121.
|- ((ph -> (ps /\ ch)) <-> ((ph -> ps) /\ (ph -> ch)))
Theorem	pm4.76 602	Theorem *4.76 of [WhiteheadRussell] p. 121.
|- (((ph -> ps) /\ (ph -> ch)) <-> (ph -> (ps /\ ch)))
Theorem	jcad 603	Deduction conjoining the consequents of two implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (ps -> th))   =>   |- (ph -> (ps -> (ch /\ th)))
Theorem	jctild 604	Deduction conjoining a theorem to left of consequent in an implication.
|- (ph -> (ps -> ch))   &   |- (ph -> th)   =>   |- (ph -> (ps -> (th /\ ch)))
Theorem	jctird 605	Deduction conjoining a theorem to right of consequent in an implication.
|- (ph -> (ps -> ch))   &   |- (ph -> th)   =>   |- (ph -> (ps -> (ch /\ th)))
Theorem	pm3.43 606	Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113.
|- (((ph -> ps) /\ (ph -> ch)) -> (ph -> (ps /\ ch)))
Theorem	andi 607	Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118.
|- ((ph /\ (ps \/ ch)) <-> ((ph /\ ps) \/ (ph /\ ch)))
Theorem	andir 608	Distributive law for conjunction.
|- (((ph \/ ps) /\ ch) <-> ((ph /\ ch) \/ (ps /\ ch)))
Theorem	orddi 609	Double distributive law for disjunction.
|- (((ph /\ ps) \/ (ch /\ th)) <-> (((ph \/ ch) /\ (ph \/ th)) /\ ((ps \/ ch) /\ (ps \/ th))))
Theorem	anddi 610	Double distributive law for conjunction.
|- (((ph \/ ps) /\ (ch \/ th)) <-> (((ph /\ ch) \/ (ph /\ th)) \/ ((ps /\ ch) \/ (ps /\ th))))
Theorem	bibi2i 611	Inference adding a biconditional to the left in an equivalence.
|- (ph <-> ps)   =>   |- ((ch <-> ph) <-> (ch <-> ps))
Theorem	bibi1i 612	Inference adding a biconditional to the right in an equivalence.
|- (ph <-> ps)   =>   |- ((ph <-> ch) <-> (ps <-> ch))
Theorem	bibi12i 613	The equivalence of two equivalences.
|- (ph <-> ps)   &   |- (ch <-> th)   =>   |- ((ph <-> ch) <-> (ps <-> th))
Theorem	notbid 614	Deduction negating both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> (-. ps <-> -. ch))
Theorem	imbi2d 615	Deduction adding an antecedent to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th -> ps) <-> (th -> ch)))
Theorem	imbi1d 616	Deduction adding a consequent to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps -> th) <-> (ch -> th)))
Theorem	orbi2d 617	Deduction adding a left disjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th \/ ps) <-> (th \/ ch)))
Theorem	orbi1d 618	Deduction adding a right disjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps \/ th) <-> (ch \/ th)))
Theorem	anbi2d 619	Deduction adding a left conjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th /\ ps) <-> (th /\ ch)))
Theorem	anbi1d 620	Deduction adding a right conjunct to both sides of a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps /\ th) <-> (ch /\ th)))
Theorem	bibi2d 621	Deduction adding a biconditional to the left in an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th <-> ps) <-> (th <-> ch)))
Theorem	bibi1d 622	Deduction adding a biconditional to the right in an equivalence.
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps <-> th) <-> (ch <-> th)))
Theorem	orbi1 623	Theorem *4.37 of [WhiteheadRussell] p. 118.
|- ((ph <-> ps) -> ((ph \/ ch) <-> (ps \/ ch)))
Theorem	anbi1 624	Theorem *4.36 of [WhiteheadRussell] p. 118.
|- ((ph <-> ps) -> ((ph /\ ch) <-> (ps /\ ch)))
Theorem	bitr 625	Theorem *4.22 of [WhiteheadRussell] p. 117.
|- (((ph <-> ps) /\ (ps <-> ch)) -> (ph <-> ch))
Theorem	imbi1 626	Theorem *4.84 of [WhiteheadRussell] p. 122.
|- ((ph <-> ps) -> ((ph -> ch) <-> (ps -> ch)))
Theorem	imbi2 627	Theorem *4.85 of [WhiteheadRussell] p. 122.
|- ((ph <-> ps) -> ((ch -> ph) <-> (ch -> ps)))
Theorem	bibi1 628	Theorem *4.86 of [WhiteheadRussell] p. 122.
|- ((ph <-> ps) -> ((ph <-> ch) <-> (ps <-> ch)))
Theorem	imbi12d 629	Deduction joining two equivalences to form equivalence of implications.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps -> th) <-> (ch -> ta)))
Theorem	orbi12d 630	Deduction joining two equivalences to form equivalence of disjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps \/ th) <-> (ch \/ ta)))
Theorem	anbi12d 631	Deduction joining two equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps /\ th) <-> (ch /\ ta)))
Theorem	bibi12d 632	Deduction joining two equivalences to form equivalence of biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps <-> th) <-> (ch <-> ta)))
Theorem	pm4.39 633	Theorem *4.39 of [WhiteheadRussell] p. 118.
|- (((ph <-> ch) /\ (ps <-> th)) -> ((ph \/ ps) <-> (ch \/ th)))
Theorem	pm4.38 634	Theorem *4.38 of [WhiteheadRussell] p. 118.
|- (((ph <-> ch) /\ (ps <-> th)) -> ((ph /\ ps) <-> (ch /\ th)))
Theorem	bi2anan9 635	Deduction joining two equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (th -> (ta <-> et))   =>   |- ((ph /\ th) -> ((ps /\ ta) <-> (ch /\ et)))
Theorem	bi2anan9r 636	Deduction joining two equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (th -> (ta <-> et))   =>   |- ((th /\ ph) -> ((ps /\ ta) <-> (ch /\ et)))
Theorem	bi2bian9 637	Deduction joining two biconditionals with different antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ta <-> et))   =>   |- ((ph /\ th) -> ((ps <-> ta) <-> (ch <-> et)))
Theorem	pm4.71 638	Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120.
|- ((ph -> ps) <-> (ph <-> (ph /\ ps)))
Theorem	pm4.71r 639	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 640	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 641	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))
Theorem	pm4.71rd 642	Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120.
|- (ph -> (ps -> ch))   =>   |- (ph -> (ps <-> (ch /\ ps)))
Theorem	pm4.45 643	Theorem *4.45 of [WhiteheadRussell] p. 119.
|- (ph <-> (ph /\ (ph \/ ps)))
Theorem	pm4.72 644	Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121.
|- ((ph -> ps) <-> (ps <-> (ph \/ ps)))
Theorem	iba 645	Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121.
|- (ph -> (ps <-> (ps /\ ph)))
Theorem	ibar 646	Introduction of antecedent as conjunct.
|- (ph -> (ps <-> (ph /\ ps)))
Theorem	pm5.32 647	Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125.
|- ((ph -> (ps <-> ch)) <-> ((ph /\ ps) <-> (ph /\ ch)))
Theorem	pm5.32i 648	Distribution of implication over biconditional (inference rule).
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ps) <-> (ph /\ ch))
Theorem	pm5.32ri 649	Distribution of implication over biconditional (inference rule).
|- (ph -> (ps <-> ch))   =>   |- ((ps /\ ph) <-> (ch /\ ph))
Theorem	pm5.32d 650	Distribution of implication over biconditional (deduction rule).
|- (ph -> (ps -> (ch <-> th)))   =>   |- (ph -> ((ps /\ ch) <-> (ps /\ th)))
Theorem	pm5.32rd 651	Distribution of implication over biconditional (deduction rule).
|- (ph -> (ps -> (ch <-> th)))   =>   |- (ph -> ((ch /\ ps) <-> (th /\ ps)))
Theorem	pm5.32da 652	Distribution of implication over biconditional (deduction rule).
|- ((ph /\ ps) -> (ch <-> th))   =>   |- (ph -> ((ps /\ ch) <-> (ps /\ th)))
Theorem	pm5.33 653	Theorem *5.33 of [WhiteheadRussell] p. 125.
|- ((ph /\ (ps -> ch)) <-> (ph /\ ((ph /\ ps) -> ch)))
Theorem	pm5.36 654	Theorem *5.36 of [WhiteheadRussell] p. 125.
|- ((ph /\ (ph <-> ps)) <-> (ps /\ (ph <-> ps)))
Theorem	pm5.42 655	Theorem *5.42 of [WhiteheadRussell] p. 125.
|- ((ph -> (ps -> ch)) <-> (ph -> (ps -> (ph /\ ch))))
Theorem	bianabs 656	Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
|- (ph -> (ps <-> (ph /\ ch)))   =>   |- (ph -> (ps <-> ch))
Theorem	oibabs 657	Absorption of disjunction into equivalence.
|- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))
Theorem	exmid 658	Law of excluded middle, also called the principle of tertium non datur. Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is either true or not true; there are no in-between values of truth. This is an essential distinction of our classical logic and is not a theorem of intuitionistic logic.
|- (ph \/ -. ph)
Theorem	pm2.1 659	Theorem *2.1 of [WhiteheadRussell] p. 101.
|- (-. ph \/ ph)
Theorem	pm2.13 660	Theorem *2.13 of [WhiteheadRussell] p. 101.
|- (ph \/ -. -. -. ph)
Theorem	pm3.24 661	Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction").
|- -. (ph /\ -. ph)
Theorem	pm2.26 662	Theorem *2.26 of [WhiteheadRussell] p. 104.
|- (-. ph \/ ((ph -> ps) -> ps))
Theorem	pm5.18 663	Theorem *5.18 of [WhiteheadRussell] p. 124. This theorem says that logical equivalence is the same as negated "exclusive-or."
|- ((ph <-> ps) <-> -. (ph <-> -. ps))
Theorem	nbbn 664	Move negation outside of biconditional. Compare Theorem *5.18 of [WhiteheadRussell] p. 124.
|- ((-. ph <-> ps) <-> -. (ph <-> ps))
Theorem	pm5.11 665	Theorem *5.11 of [WhiteheadRussell] p. 123.
|- ((ph -> ps) \/ (-. ph -> ps))
Theorem	pm5.12 666	Theorem *5.12 of [WhiteheadRussell] p. 123.
|- ((ph -> ps) \/ (ph -> -. ps))
Theorem	pm5.13 667	Theorem *5.13 of [WhiteheadRussell] p. 123.
|- ((ph -> ps) \/ (ps -> ph))
Theorem	pm5.14 668	Theorem *5.14 of [WhiteheadRussell] p. 123.
|- ((ph -> ps) \/ (ps -> ch))
Theorem	pm5.15 669	Theorem *5.15 of [WhiteheadRussell] p. 124.
|- ((ph <-> ps) \/ (ph <-> -. ps))
Theorem	pm5.16 670	Theorem *5.16 of [WhiteheadRussell] p. 124.
|- -. ((ph <-> ps) /\ (ph <-> -. ps))
Theorem	pm5.17 671	Theorem *5.17 of [WhiteheadRussell] p. 124.
|- (((ph \/ ps) /\ -. (ph /\ ps)) <-> (ph <-> -. ps))
Theorem	pm5.19 672	Theorem *5.19 of [WhiteheadRussell] p. 124.
|- -. (ph <-> -. ph)
Theorem	dfbi3 673	An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124.
|- ((ph <-> ps) <-> ((ph /\ ps) \/ (-. ph /\ -. ps)))
Theorem	xor 674	Two ways to express "exclusive or." Theorem *5.22 of [WhiteheadRussell] p. 124.
|- (-. (ph <-> ps) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))
Theorem	pm5.24 675	Theorem *5.24 of [WhiteheadRussell] p. 124.
|- (-. ((ph /\ ps) \/ (-. ph /\ -. ps)) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))
Theorem	xor2 676	Two ways to express "exclusive or."
|- (-. (ph <-> ps) <-> ((ph \/ ps) /\ -. (ph /\ ps)))
Theorem	xor3 677	Two ways to express "exclusive or."
|- (-. (ph <-> ps) <-> (ph <-> -. ps))
Theorem	xordi 678	Conjunction distributes over exclusive-or, using -. (ph <-> ps) to express exclusive-or. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic.
|- ((ph /\ -. (ps <-> ch)) <-> -. ((ph /\ ps) <-> (ph /\ ch)))
Theorem	pm5.55 679	Theorem *5.55 of [WhiteheadRussell] p. 125.
|- (((ph \/ ps) <-> ph) \/ ((ph \/ ps) <-> ps))
Miscellaneous theorems of propositional calculus
Theorem	pm5.1 680	Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123.
|- ((ph /\ ps) -> (ph <-> ps))
Theorem	pm5.21 681	Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124.
|- ((-. ph /\ -. ps) -> (ph <-> ps))
Theorem	pm5.21ni 682	Two propositions implying a false one are equivalent.
|- (ph -> ps)   &   |- (ch -> ps)   =>   |- (-. ps -> (ph <-> ch))
Theorem	pm5.21nii 683	Eliminate an antecedent implied by each side of a biconditional.
|- (ph -> ps)   &   |- (ch -> ps)   &   |- (ps -> (ph <-> ch))   =>   |- (ph <-> ch)
Theorem	pm5.21nd 684	Eliminate an antecedent implied by each side of a biconditional.
|- ((ph /\ ps) -> th)   &   |- ((ph /\ ch) -> th)   &   |- (th -> (ps <-> ch))   =>   |- (ph -> (ps <-> ch))
Theorem	bibif 685	Transfer negation via an equivalence.
|- (-. ps -> ((ph <-> ps) <-> -. ph))
Theorem	pm5.35 686	Theorem *5.35 of [WhiteheadRussell] p. 125.
|- (((ph -> ps) /\ (ph -> ch)) -> (ph -> (ps <-> ch)))
Theorem	pm5.54 687	Theorem *5.54 of [WhiteheadRussell] p. 125.
|- (((ph /\ ps) <-> ph) \/ ((ph /\ ps) <-> ps))
Theorem	elimant 688	Elimination of antecedents in an implication. (The proof was shortened by Juha Arpiainen, 19-Jan-2006.)
|- (((ph -> ps) /\ ((ps -> ch) -> (ph -> th))) -> (ph -> (ch -> th)))
Theorem	baib 689	Move conjunction outside of biconditional.
|- (ph <-> (ps /\ ch))   =>   |- (ps -> (ph <-> ch))
Theorem	baibr 690	Move conjunction outside of biconditional.
|- (ph <-> (ps /\ ch))   =>   |- (ps -> (ch <-> ph))
Theorem	pm5.44 691	Theorem *5.44 of [WhiteheadRussell] p. 125.
|- ((ph -> ps) -> ((ph -> ch) <-> (ph -> (ps /\ ch))))
Theorem	pm5.6 692	Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125.
|- (((ph /\ -. ps) -> ch) <-> (ph -> (ps \/ ch)))
Theorem	nan 693	Theorem to move a conjunct in and out of a negation.
|- ((ph -> -. (ps /\ ch)) <-> ((ph /\ ps) -> -. ch))
Theorem	orcanai 694	Change disjunction in consequent to conjunction in antecedent.
|- (ph -> (ps \/ ch))   =>   |- ((ph /\ -. ps) -> ch)
Theorem	intnan 695	Introduction of conjunct inside of a contradiction.
|- -. ph   =>   |- -. (ps /\ ph)
Theorem	intnanr 696	Introduction of conjunct inside of a contradiction.
|- -. ph   =>   |- -. (ph /\ ps)
Theorem	intnand 697	Introduction of conjunct inside of a contradiction.
|- (ph -> -. ps)   =>   |- (ph -> -. (ch /\ ps))
Theorem	intnanrd 698	Introduction of conjunct inside of a contradiction.
|- (ph -> -. ps)   =>   |- (ph -> -. (ps /\ ch))
Theorem	mpan 699	An inference based on modus ponens.
|- ph   &   |- ((ph /\ ps) -> ch)   =>   |- (ps -> ch)
Theorem	mpan2 700	An inference based on modus ponens.
|- ps   &   |- ((ph /\ ps) -> ch)   =>   |- (ph -> ch)