mmtheorems8 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 701-800 - Page 8 of 107

Type	Label	Description
Statement
Theorem	mpan2d 701	A deduction based on modus ponens.
|- (ph -> ch)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ps -> th))
Theorem	mp2and 702	A deduction based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> th)
Theorem	mpdan 703	An inference based on modus ponens.
|- (ph -> ps)   &   |- ((ph /\ ps) -> ch)   =>   |- (ph -> ch)
Theorem	mpancom 704	An inference based on modus ponens with commutation of antecedents.
|- (ps -> ph)   &   |- ((ph /\ ps) -> ch)   =>   |- (ps -> ch)
Theorem	mpanl1 705	An inference based on modus ponens.
|- ph   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ps /\ ch) -> th)
Theorem	mpanl2 706	An inference based on modus ponens.
|- ps   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mpanl12 707	An inference based on modus ponens.
|- ph   &   |- ps   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- (ch -> th)
Theorem	mpanr1 708	An inference based on modus ponens.
|- ps   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mpanr2 709	An inference based on modus ponens.
|- ch   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpanlr1 710	An inference based on modus ponens.
|- ps   &   |- (((ph /\ (ps /\ ch)) /\ th) -> ta)   =>   |- (((ph /\ ch) /\ th) -> ta)
Theorem	mtt 711	Modus-tollens-like theorem.
|- (-. ph -> (-. ps <-> (ps -> ph)))
Theorem	mt2bi 712	A false consequent falsifies an antecedent.
|- ph   =>   |- (-. ps <-> (ps -> -. ph))
Theorem	mtbid 713	A deduction from a biconditional, similar to modus tollens.
|- (ph -> -. ps)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ch)
Theorem	mtbird 714	A deduction from a biconditional, similar to modus tollens.
|- (ph -> -. ch)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ps)
Theorem	mtbii 715	An inference from a biconditional, similar to modus tollens.
|- -. ps   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ch)
Theorem	mtbiri 716	An inference from a biconditional, similar to modus tollens.
|- -. ch   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ps)
Theorem	2th 717	Two truths are equivalent.
|- ph   &   |- ps   =>   |- (ph <-> ps)
Theorem	2false 718	Two falsehoods are equivalent.
|- -. ph   &   |- -. ps   =>   |- (ph <-> ps)
Theorem	tbt 719	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 720	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 721	The negation of a wff is equivalent to the wff's equivalence to falsehood.
|- -. ph   =>   |- (-. ps <-> (ps <-> ph))
Theorem	nbn3 722	Transfer falsehood via equivalence.
|- ph   =>   |- (-. ps <-> (ps <-> -. ph))
Theorem	biantru 723	A wff is equivalent to its conjunction with truth.
|- ph   =>   |- (ps <-> (ps /\ ph))
Theorem	biantrur 724	A wff is equivalent to its conjunction with truth.
|- ph   =>   |- (ps <-> (ph /\ ps))
Theorem	biantrud 725	A wff is equivalent to its conjunction with truth.
|- (ph -> ps)   =>   |- (ph -> (ch <-> (ch /\ ps)))
Theorem	biantrurd 726	A wff is equivalent to its conjunction with truth.
|- (ph -> ps)   =>   |- (ph -> (ch <-> (ps /\ ch)))
Theorem	mpbiran 727	Detach truth from conjunction in biconditional.
|- (ph <-> (ps /\ ch))   &   |- ps   =>   |- (ph <-> ch)
Theorem	mpbiran2 728	Detach truth from conjunction in biconditional.
|- (ph <-> (ps /\ ch))   &   |- ch   =>   |- (ph <-> ps)
Theorem	mpbir2an 729	Detach a conjunction of truths in a biconditional.
|- (ph <-> (ps /\ ch))   &   |- ps   &   |- ch   =>   |- ph
Theorem	biimt 730	A wff is equivalent to itself with true antecedent.
|- (ph -> (ps <-> (ph -> ps)))
Theorem	pm5.5 731	Theorem *5.5 of [WhiteheadRussell] p. 125.
|- (ph -> ((ph -> ps) <-> ps))
Theorem	pm5.62 732	Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 21-Jun-2005.)
|- (((ph /\ ps) \/ -. ps) <-> (ph \/ -. ps))
Theorem	biort 733	A wff is disjoined with truth is true.
|- (ph -> (ph <-> (ph \/ ps)))
Theorem	biorf 734	A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121.
|- (-. ph -> (ps <-> (ph \/ ps)))
Theorem	biorfi 735	A wff is equivalent to its disjunction with falsehood.
|- -. ph   =>   |- (ps <-> (ps \/ ph))
Theorem	bianfi 736	A wff conjoined with falsehood is false.
|- -. ph   =>   |- (ph <-> (ps /\ ph))
Theorem	bianfd 737	A wff conjoined with falsehood is false.
|- (ph -> -. ps)   =>   |- (ph -> (ps <-> (ps /\ ch)))
Theorem	pm4.82 738	Theorem *4.82 of [WhiteheadRussell] p. 122.
|- (((ph -> ps) /\ (ph -> -. ps)) <-> -. ph)
Theorem	pm4.83 739	Theorem *4.83 of [WhiteheadRussell] p. 122.
|- (((ph -> ps) /\ (-. ph -> ps)) <-> ps)
Theorem	pclem6 740	Negation inferred from embedded conjunct.
|- ((ph <-> (ps /\ -. ph)) -> -. ps)
Theorem	biantr 741	A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117.
|- (((ph <-> ps) /\ (ch <-> ps)) -> (ph <-> ch))
Theorem	orbidi 742	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 743	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 744	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	pm5.7 745	Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 742. (Contributed by Roy F. Longton, 21-Jun-2005.)
|- (((ph \/ ch) <-> (ps \/ ch)) <-> (ch \/ (ph <-> ps)))
Theorem	bigolden 746	Dijkstra-Scholten's Golden Rule for calculational proofs.
|- (((ph /\ ps) <-> ph) <-> (ps <-> (ph \/ ps)))
Theorem	pm5.71 747	Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.)
|- ((ps -> -. ch) -> (((ph \/ ps) /\ ch) <-> (ph /\ ch)))
Theorem	pm5.75 748	Theorem *5.75 of [WhiteheadRussell] p. 126.
|- (((ch -> -. ps) /\ (ph <-> (ps \/ ch))) -> ((ph /\ -. ps) <-> ch))
Theorem	bimsc1 749	Removal of conjunct from one side of an equivalence.
|- (((ph -> ps) /\ (ch <-> (ps /\ ph))) -> (ch <-> ph))
Theorem	ecase2d 750	Deduction for elimination by cases.
|- (ph -> ps)   &   |- (ph -> -. (ps /\ ch))   &   |- (ph -> -. (ps /\ th))   &   |- (ph -> (ta \/ (ch \/ th)))   =>   |- (ph -> ta)
Theorem	ecase3 751	Inference for elimination by cases.
|- (ph -> ch)   &   |- (ps -> ch)   &   |- (-. (ph \/ ps) -> ch)   =>   |- ch
Theorem	ecase 752	Inference for elimination by cases.
|- (-. ph -> ch)   &   |- (-. ps -> ch)   &   |- ((ph /\ ps) -> ch)   =>   |- ch
Theorem	ecase3d 753	Deduction for elimination by cases.
|- (ph -> (ps -> th))   &   |- (ph -> (ch -> th))   &   |- (ph -> (-. (ps \/ ch) -> th))   =>   |- (ph -> th)
Theorem	ccase 754	Inference for combining cases.
|- ((ph /\ ps) -> ta)   &   |- ((ch /\ ps) -> ta)   &   |- ((ph /\ th) -> ta)   &   |- ((ch /\ th) -> ta)   =>   |- (((ph \/ ch) /\ (ps \/ th)) -> ta)
Theorem	ccased 755	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 756	Inference for combining cases.
|- ((ph /\ ps) -> ta)   &   |- (ch -> ta)   &   |- (th -> ta)   =>   |- (((ph \/ ch) /\ (ps \/ th)) -> ta)
Theorem	4cases 757	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 758	Miscellaneous inference relating falsehoods.
|- ph   =>   |- (-. ps -> ((ch /\ ps) <-> -. ph))
Theorem	dedlem0a 759	Lemma for an alternate version of weak deduction theorem.
Theorem	dedlem0b 760	Lemma for an alternate version of weak deduction theorem.
Theorem	dedlema 761	Lemma for weak deduction theorem.
Theorem	dedlemb 762	Lemma for weak deduction theorem.
Theorem	elimh 763	Hypothesis builder for weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page.
|- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (ch <-> ta))   &   |- ((ps <->