mmtheorems8 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 701-800 - Page 8 of 108

Type	Label	Description
Statement
Theorem	mpand 701	A deduction based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ch -> th))
Theorem	mpan2d 702	A deduction based on modus ponens.
|- (ph -> ch)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ps -> th))
Theorem	mp2and 703	A deduction based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> th)
Theorem	mpdan 704	An inference based on modus ponens.
|- (ph -> ps)   &   |- ((ph /\ ps) -> ch)   =>   |- (ph -> ch)
Theorem	mpancom 705	An inference based on modus ponens with commutation of antecedents.
|- (ps -> ph)   &   |- ((ph /\ ps) -> ch)   =>   |- (ps -> ch)
Theorem	mpanl1 706	An inference based on modus ponens.
|- ph   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ps /\ ch) -> th)
Theorem	mpanl2 707	An inference based on modus ponens.
|- ps   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mpanl12 708	An inference based on modus ponens.
|- ph   &   |- ps   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- (ch -> th)
Theorem	mpanr1 709	An inference based on modus ponens.
|- ps   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mpanr2 710	An inference based on modus ponens.
|- ch   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpanlr1 711	An inference based on modus ponens.
|- ps   &   |- (((ph /\ (ps /\ ch)) /\ th) -> ta)   =>   |- (((ph /\ ch) /\ th) -> ta)
Theorem	mtt 712	Modus-tollens-like theorem.
|- (-. ph -> (-. ps <-> (ps -> ph)))
Theorem	mt2bi 713	A false consequent falsifies an antecedent.
|- ph   =>   |- (-. ps <-> (ps -> -. ph))
Theorem	mtbid 714	A deduction from a biconditional, similar to modus tollens.
|- (ph -> -. ps)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ch)
Theorem	mtbird 715	A deduction from a biconditional, similar to modus tollens.
|- (ph -> -. ch)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ps)
Theorem	mtbii 716	An inference from a biconditional, similar to modus tollens.
|- -. ps   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ch)
Theorem	mtbiri 717	An inference from a biconditional, similar to modus tollens.
|- -. ch   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ps)
Theorem	2th 718	Two truths are equivalent.
|- ph   &   |- ps   =>   |- (ph <-> ps)
Theorem	2false 719	Two falsehoods are equivalent.
|- -. ph   &   |- -. ps   =>   |- (ph <-> ps)
Theorem	tbt 720	A wff is equivalent to its equivalence with truth. (The proof was shortened by Juha Arpiainen, 19-Jan-2006.)
|- ph   =>   |- (ps <-> (ps <-> ph))
Theorem	nbn2 721	The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.)
|- (-. ph -> (-. ps <-> (ph <-> ps)))
Theorem	nbn 722	The negation of a wff is equivalent to the wff's equivalence to falsehood.
|- -. ph   =>   |- (-. ps <-> (ps <-> ph))
Theorem	nbn3 723	Transfer falsehood via equivalence.
|- ph   =>   |- (-. ps <-> (ps <-> -. ph))
Theorem	biantru 724	A wff is equivalent to its conjunction with truth.
|- ph   =>   |- (ps <-> (ps /\ ph))
Theorem	biantrur 725	A wff is equivalent to its conjunction with truth.
|- ph   =>   |- (ps <-> (ph /\ ps))
Theorem	biantrud 726	A wff is equivalent to its conjunction with truth.
|- (ph -> ps)   =>   |- (ph -> (ch <-> (ch /\ ps)))
Theorem	biantrurd 727	A wff is equivalent to its conjunction with truth.
|- (ph -> ps)   =>   |- (ph -> (ch <-> (ps /\ ch)))
Theorem	mpbiran 728	Detach truth from conjunction in biconditional.
|- (ph <-> (ps /\ ch))   &   |- ps   =>   |- (ph <-> ch)
Theorem	mpbiran2 729	Detach truth from conjunction in biconditional.
|- (ph <-> (ps /\ ch))   &   |- ch   =>   |- (ph <-> ps)
Theorem	mpbir2an 730	Detach a conjunction of truths in a biconditional.
|- (ph <-> (ps /\ ch))   &   |- ps   &   |- ch   =>   |- ph
Theorem	biimt 731	A wff is equivalent to itself with true antecedent.
|- (ph -> (ps <-> (ph -> ps)))
Theorem	pm5.5 732	Theorem *5.5 of [WhiteheadRussell] p. 125.
|- (ph -> ((ph -> ps) <-> ps))
Theorem	pm5.62 733	Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 21-Jun-2005.)
|- (((ph /\ ps) \/ -. ps) <-> (ph \/ -. ps))
Theorem	biort 734	A wff is disjoined with truth is true.
|- (ph -> (ph <-> (ph \/ ps)))
Theorem	biorf 735	A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121.
|- (-. ph -> (ps <-> (ph \/ ps)))
Theorem	biorfi 736	A wff is equivalent to its disjunction with falsehood.
|- -. ph   =>   |- (ps <-> (ps \/ ph))
Theorem	bianfi 737	A wff conjoined with falsehood is false.
|- -. ph   =>   |- (ph <-> (ps /\ ph))
Theorem	bianfd 738	A wff conjoined with falsehood is false.
|- (ph -> -. ps)   =>   |- (ph -> (ps <-> (ps /\ ch)))
Theorem	pm4.82 739	Theorem *4.82 of [WhiteheadRussell] p. 122.
|- (((ph -> ps) /\ (ph -> -. ps)) <-> -. ph)
Theorem	pm4.83 740	Theorem *4.83 of [WhiteheadRussell] p. 122.
|- (((ph -> ps) /\ (-. ph -> ps)) <-> ps)
Theorem	pclem6 741	Negation inferred from embedded conjunct.
|- ((ph <-> (ps /\ -. ph)) -> -. ps)
Theorem	biantr 742	A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117.
|- (((ph <-> ps) /\ (ch <-> ps)) -> (ph <-> ch))
Theorem	orbidi 743	Disjunction distributes over the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseer.ist.psu.edu/lifschitz98calculational.html.
|- ((ph \/ (ps <-> ch)) <-> ((ph \/ ps) <-> (ph \/ ch)))
Theorem	biass 744	Associative law for the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseer.ist.psu.edu/lifschitz98calculational.html. Interestingly, this law was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (The proof was shortened by Juha Arpiainen, 19-Jan-2006.)
|- (((ph <-> ps) <-> ch) <-> (ph <-> (ps <-> ch)))
Theorem	biluk 745	Lukasiewicz's shortest axiom for equivalential calculus. Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96.
|- ((ph <-> ps) <-> ((ch <-> ps) <-> (ph <-> ch)))
Theorem	pm5.7 746	Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 743. (Contributed by Roy F. Longton, 21-Jun-2005.)
|- (((ph \/ ch) <-> (ps \/ ch)) <-> (ch \/ (ph <-> ps)))
Theorem	bigolden 747	Dijkstra-Scholten's Golden Rule for calculational proofs.
|- (((ph /\ ps) <-> ph) <-> (ps <-> (ph \/ ps)))
Theorem	pm5.71 748	Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.)
|- ((ps -> -. ch) -> (((ph \/ ps) /\ ch) <-> (ph /\ ch)))
Theorem	pm5.75 749	Theorem *5.75 of [WhiteheadRussell] p. 126.
|- (((ch -> -. ps) /\ (ph <-> (ps \/ ch))) -> ((ph /\ -. ps) <-> ch))
Theorem	bimsc1 750	Removal of conjunct from one side of an equivalence.
|- (((ph -> ps) /\ (ch <-> (ps /\ ph))) -> (ch <-> ph))
Theorem	ecase2d 751	Deduction for elimination by cases.
|- (ph -> ps)   &   |- (ph -> -. (ps /\ ch))   &   |- (ph -> -. (ps /\ th))   &   |- (ph -> (ta \/ (ch \/ th)))   =>   |- (ph -> ta)
Theorem	ecase3 752	Inference for elimination by cases.
|- (ph -> ch)   &   |- (ps -> ch)   &   |- (-. (ph \/ ps) -> ch)   =>   |- ch
Theorem	ecase 753	Inference for elimination by cases.
|- (-. ph -> ch)   &   |- (-. ps -> ch)   &   |- ((ph /\ ps) -> ch)   =>   |- ch
Theorem	ecase3d 754	Deduction for elimination by cases.
|- (ph -> (ps -> th))   &   |- (ph -> (ch -> th))   &   |- (ph -> (-. (ps \/ ch) -> th))   =>   |- (ph -> th)
Theorem	ccase 755	Inference for combining cases.
|- ((ph /\ ps) -> ta)   &   |- ((ch /\ ps) -> ta)   &   |- ((ph /\ th) -> ta)   &   |- ((ch /\ th) -> ta)   =>   |- (((ph \/ ch) /\ (ps \/ th)) -> ta)
Theorem	ccased 756	Deduction for combining cases.
|- (ph -> ((ps /\ ch) -> et))   &   |- (ph -> ((th /\ ch) -> et))   &   |- (ph -> ((ps /\ ta) -> et))   &   |- (ph -> ((th /\ ta) -> et))   =>   |- (ph -> (((ps \/ th) /\ (ch \/ ta)) -> et))
Theorem	ccase2 757	Inference for combining cases.
|- ((ph /\ ps) -> ta)   &   |- (ch -> ta)   &   |- (th -> ta)   =>   |- (((ph \/ ch) /\ (ps \/ th)) -> ta)
Theorem	4cases 758	Inference eliminating two antecedents from the four possible cases that result from their true/false combinations.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ -. ps) -> ch)   &   |- ((-. ph /\ ps) -> ch)   &   |- ((-. ph /\ -. ps) -> ch)   =>   |- ch
Theorem	niabn 759	Miscellaneous inference relating falsehoods.
|- ph   =>   |- (-. ps -> ((ch /\ ps) <-> -. ph))
Theorem	dedlem0a 760	Lemma for an alternate version of weak deduction theorem.
Theorem	dedlem0b 761	Lemma for an alternate version of weak deduction theorem.
Theorem	dedlema 762	Lemma for weak deduction theorem.
Theorem	dedlemb 763	Lemma for weak deduction theorem.