mmtheorems8 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 701-800 - Page 8 of 175

Type	Label	Description
Statement
Theorem	pm4.71rd 701	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	pm4.45 702	Theorem *4.45 of [WhiteheadRussell] p. 119.
|- (ph <-> (ph /\ (ph \/ ps)))
Theorem	pm4.72 703	Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121.
|- ((ph -> ps) <-> (ps <-> (ph \/ ps)))
Theorem	iba 704	Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121.
|- (ph -> (ps <-> (ps /\ ph)))
Theorem	ibar 705	Introduction of antecedent as conjunct.
|- (ph -> (ps <-> (ph /\ ps)))
Theorem	pm5.32 706	Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125.
|- ((ph -> (ps <-> ch)) <-> ((ph /\ ps) <-> (ph /\ ch)))
Theorem	pm5.32i 707	Distribution of implication over biconditional (inference rule).
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ps) <-> (ph /\ ch))
Theorem	pm5.32ri 708	Distribution of implication over biconditional (inference rule).
|- (ph -> (ps <-> ch))   =>   |- ((ps /\ ph) <-> (ch /\ ph))
Theorem	pm5.32d 709	Distribution of implication over biconditional (deduction rule).
|- (ph -> (ps -> (ch <-> th)))   =>   |- (ph -> ((ps /\ ch) <-> (ps /\ th)))
Theorem	pm5.32rd 710	Distribution of implication over biconditional (deduction rule).
|- (ph -> (ps -> (ch <-> th)))   =>   |- (ph -> ((ch /\ ps) <-> (th /\ ps)))
Theorem	pm5.32da 711	Distribution of implication over biconditional (deduction rule).
|- ((ph /\ ps) -> (ch <-> th))   =>   |- (ph -> ((ps /\ ch) <-> (ps /\ th)))
Theorem	pm5.33 712	Theorem *5.33 of [WhiteheadRussell] p. 125.
|- ((ph /\ (ps -> ch)) <-> (ph /\ ((ph /\ ps) -> ch)))
Theorem	pm5.36 713	Theorem *5.36 of [WhiteheadRussell] p. 125.
|- ((ph /\ (ph <-> ps)) <-> (ps /\ (ph <-> ps)))
Theorem	pm5.42 714	Theorem *5.42 of [WhiteheadRussell] p. 125.
|- ((ph -> (ps -> ch)) <-> (ph -> (ps -> (ph /\ ch))))
Theorem	bianabs 715	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 716	Absorption of disjunction into equivalence.
|- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))
Theorem	exmid 717	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	pm2.1 718	Theorem *2.1 of [WhiteheadRussell] p. 101.
|- (-. ph \/ ph)
Theorem	pm2.13 719	Theorem *2.13 of [WhiteheadRussell] p. 101.
|- (ph \/ -. -. -. ph)
Theorem	pm3.24 720	Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction").
|- -. (ph /\ -. ph)
Theorem	pm2.26 721	Theorem *2.26 of [WhiteheadRussell] p. 104.
|- (-. ph \/ ((ph -> ps) -> ps))
Theorem	pm5.18 722	Theorem *5.18 of [WhiteheadRussell] p. 124. This theorem says that logical equivalence is the same as negated "exclusive-or." (The proof was shortened by Andrew Salmon, 20-Jun-2011.)
|- ((ph <-> ps) <-> -. (ph <-> -. ps))
Theorem	pm5.18OLD 723	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 724	Move negation outside of biconditional. Compare Theorem *5.18 of [WhiteheadRussell] p. 124.
|- ((-. ph <-> ps) <-> -. (ph <-> ps))
Theorem	pm5.11 725	Theorem *5.11 of [WhiteheadRussell] p. 123.
|- ((ph -> ps) \/ (-. ph -> ps))
Theorem	pm5.12 726	Theorem *5.12 of [WhiteheadRussell] p. 123.
|- ((ph -> ps) \/ (ph -> -. ps))
Theorem	pm5.13 727	Theorem *5.13 of [WhiteheadRussell] p. 123.
|- ((ph -> ps) \/ (ps -> ph))
Theorem	pm5.14 728	Theorem *5.14 of [WhiteheadRussell] p. 123.
|- ((ph -> ps) \/ (ps -> ch))
Theorem	pm5.15 729	Theorem *5.15 of [WhiteheadRussell] p. 124.
|- ((ph <-> ps) \/ (ph <-> -. ps))
Theorem	pm5.16 730	Theorem *5.16 of [WhiteheadRussell] p. 124.
|- -. ((ph <-> ps) /\ (ph <-> -. ps))
Theorem	pm5.17 731	Theorem *5.17 of [WhiteheadRussell] p. 124.
|- (((ph \/ ps) /\ -. (ph /\ ps)) <-> (ph <-> -. ps))
Theorem	pm5.19 732	Theorem *5.19 of [WhiteheadRussell] p. 124.
|- -. (ph <-> -. ph)
Theorem	dfbi3 733	An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124.
|- ((ph <-> ps) <-> ((ph /\ ps) \/ (-. ph /\ -. ps)))
Theorem	xor 734	Two ways to express "exclusive or." Theorem *5.22 of [WhiteheadRussell] p. 124.
|- (-. (ph <-> ps) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))
Theorem	pm5.24 735	Theorem *5.24 of [WhiteheadRussell] p. 124.
|- (-. ((ph /\ ps) \/ (-. ph /\ -. ps)) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))
Theorem	xor2 736	Two ways to express "exclusive or."
|- (-. (ph <-> ps) <-> ((ph \/ ps) /\ -. (ph /\ ps)))
Theorem	xor3 737	Two ways to express "exclusive or."
|- (-. (ph <-> ps) <-> (ph <-> -. ps))
Theorem	xordi 738	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	pm5.55 739	Theorem *5.55 of [WhiteheadRussell] p. 125.
|- (((ph \/ ps) <-> ph) \/ ((ph \/ ps) <-> ps))
Miscellaneous theorems of propositional calculus
Theorem	pm5.1 740	Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123.
|- ((ph /\ ps) -> (ph <-> ps))
Theorem	pm5.21 741	Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124.
|- ((-. ph /\ -. ps) -> (ph <-> ps))
Theorem	pm5.21ni 742	Two propositions implying a false one are equivalent.
|- (ph -> ps)   &   |- (ch -> ps)   =>   |- (-. ps -> (ph <-> ch))
Theorem	pm5.21nii 743	Eliminate an antecedent implied by each side of a biconditional.
|- (ph -> ps)   &   |- (ch -> ps)   &   |- (ps -> (ph <-> ch))   =>   |- (ph <-> ch)
Theorem	pm5.21nd 744	Eliminate an antecedent implied by each side of a biconditional.
|- ((ph /\ ps) -> th)   &   |- ((ph /\ ch) -> th)   &   |- (th -> (ps <-> ch))   =>   |- (ph -> (ps <-> ch))
Theorem	bibif 745	Transfer negation via an equivalence.
|- (-. ps -> ((ph <-> ps) <-> -. ph))
Theorem	pm5.35 746	Theorem *5.35 of [WhiteheadRussell] p. 125.
|- (((ph -> ps) /\ (ph -> ch)) -> (ph -> (ps <-> ch)))
Theorem	pm5.54 747	Theorem *5.54 of [WhiteheadRussell] p. 125.
|- (((ph /\ ps) <-> ph) \/ ((ph /\ ps) <-> ps))
Theorem	elimant 748	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 749	Move conjunction outside of biconditional.
|- (ph <-> (ps /\ ch))   =>   |- (ps -> (ph <-> ch))
Theorem	baibr 750	Move conjunction outside of biconditional.
|- (ph <-> (ps /\ ch))   =>   |- (ps -> (ch <-> ph))
Theorem	pm5.44 751	Theorem *5.44 of [WhiteheadRussell] p. 125.
|- ((ph -> ps) -> ((ph -> ch) <-> (ph -> (ps /\ ch))))
Theorem	pm5.6 752	Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125.
|- (((ph /\ -. ps) -> ch) <-> (ph -> (ps \/ ch)))
Theorem	nan 753	Theorem to move a conjunct in and out of a negation.
|- ((ph -> -. (ps /\ ch)) <-> ((ph /\ ps) -> -. ch))
Theorem	orcanai 754	Change disjunction in consequent to conjunction in antecedent.
|- (ph -> (ps \/ ch))   =>   |- ((ph /\ -. ps) -> ch)
Theorem	intnan 755	Introduction of conjunct inside of a contradiction.
|- -. ph   =>   |- -. (ps /\ ph)
Theorem	intnanr 756	Introduction of conjunct inside of a contradiction.
|- -. ph   =>   |- -. (ph /\ ps)
Theorem	intnand 757	Introduction of conjunct inside of a contradiction.
|- (ph -> -. ps)   =>   |- (ph -> -. (ch /\ ps))
Theorem	intnanrd 758	Introduction of conjunct inside of a contradiction.
|- (ph -> -. ps)   =>   |- (ph -> -. (ps /\ ch))
Theorem	mpan 759	An inference based on modus ponens.
|- ph   &   |- ((ph /\ ps) -> ch)   =>   |- (ps -> ch)
Theorem	mpan2 760	An inference based on modus ponens.
|- ps   &   |- ((ph /\ ps) -> ch)   =>   |- (ph -> ch)
Theorem	mp2an 761	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ((ph /\ ps) -> ch)   =>   |- ch
Theorem	mpani 762	An inference based on modus ponens.
|- ps   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ch -> th))
Theorem	mpan2i 763	An inference based on modus ponens.
|- ch   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ps -> th))
Theorem	mp2ani 764	An inference based on modus ponens.
|- ps   &   |- ch   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> th)
Theorem	mpand 765	A deduction based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ch -> th))
Theorem	mpan2d 766	A deduction based on modus ponens.
|- (ph -> ch)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ps -> th))
Theorem	mp2and 767	A deduction based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> th)
Theorem	mpdan 768	An inference based on modus ponens.
|- (ph -> ps)   &   |- ((ph /\ ps) -> ch)   =>   |- (ph -> ch)
Theorem	mpancom 769	An inference based on modus ponens with commutation of antecedents.
|- (ps -> ph)   &   |- ((ph /\ ps) -> ch)   =>   |- (ps -> ch)
Theorem	mpanl1 770	An inference based on modus ponens.
|- ph   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ps /\ ch) -> th)
Theorem	mpanl2 771	An inference based on modus ponens. (The proof was shortened by Andrew Salmon, 7-May-2011.)
|- ps   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mpanl2OLD 772	An inference based on modus ponens.
|- ps   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mpanl12 773	An inference based on modus ponens.
|- ph   &   |- ps   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- (ch -> th)
Theorem	mpanr1 774	An inference based on modus ponens. (The proof was shortened by Andrew Salmon, 7-May-2011.)
|- ps   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mpanr1OLD 775	An inference based on modus ponens.
|- ps   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mpanr2 776	An inference based on modus ponens. (The proof was shortened by Andrew Salmon, 7-May-2011.)
|- ch   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpanr2OLD 777	An inference based on modus ponens.
|- ch   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpanr12 778	An inference based on modus ponens.
|- ps   &   |- ch   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- (ph -> th)
Theorem	mpanlr1 779	An inference based on modus ponens.
|- ps   &   |- (((ph /\ (ps /\ ch)) /\ th) -> ta)   =>   |- (((ph /\ ch) /\ th) -> ta)
Theorem	mtt 780	Modus-tollens-like theorem.
|- (-. ph -> (-. ps <-> (ps -> ph)))
Theorem	mt2bi 781	A false consequent falsifies an antecedent.
|- ph   =>   |- (-. ps <-> (ps -> -. ph))
Theorem	mtbid 782	A deduction from a biconditional, similar to modus tollens.
|- (ph -> -. ps)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ch)
Theorem	mtbird 783	A deduction from a biconditional, similar to modus tollens.
|- (ph -> -. ch)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ps)
Theorem	mtbii 784	An inference from a biconditional, similar to modus tollens.
|- -. ps   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ch)
Theorem	mtbiri 785	An inference from a biconditional, similar to modus tollens.
|- -. ch   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ps)
Theorem	2th 786	Two truths are equivalent.
|- ph   &   |- ps   =>   |- (ph <-> ps)
Theorem	2false 787	Two falsehoods are equivalent.
|- -. ph   &   |- -. ps   =>   |- (ph <-> ps)
Theorem	tbt 788	A wff is equivalent to its equivalence with truth. (The proof was shortened by Andrew Salmon, 13-May-2011.)
|- ph   =>   |- (ps <-> (ps <-> ph))
Theorem	tbtOLD 789	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 790	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 791	The negation of a wff is equivalent to the wff's equivalence to falsehood.
|- -. ph   =>   |- (-. ps <-> (ps <-> ph))
Theorem	nbn3 792	Transfer falsehood via equivalence.
|- ph   =>   |- (-. ps <-> (ps <-> -. ph))
Theorem	biantru 793	A wff is equivalent to its conjunction with truth.
|- ph   =>   |- (ps <-> (ps /\ ph))
Theorem	biantrur 794	A wff is equivalent to its conjunction with truth.
|- ph   =>   |- (ps <-> (ph /\ ps))
Theorem	biantrud 795	A wff is equivalent to its conjunction with truth.
|- (ph -> ps)   =>   |- (ph -> (ch <-> (ch /\ ps)))
Theorem	biantrurd 796	A wff is equivalent to its conjunction with truth. (The proof was shortened by Andrew Salmon, 7-May-2011.)
|- (ph -> ps)   =>   |- (ph -> (ch <-> (ps /\ ch)))
Theorem	biantrurdOLD 797	A wff is equivalent to its conjunction with truth.
|- (ph -> ps)   =>   |- (ph -> (ch <-> (ps /\ ch)))
Theorem	mpbiran 798	Detach truth from conjunction in biconditional.
|- (ph <-> (ps /\ ch))   &   |- ps   =>   |- (ph <-> ch)
Theorem	mpbiran2 799	Detach truth from conjunction in biconditional.
|- (ph <-> (ps /\ ch))   &   |- ch   =>   |- (ph <-> ps)
Theorem	mpbir2an 800	Detach a conjunction of truths in a biconditional.
|- (ph <-> (ps /\ ch))   &   |- ps   &   |- ch   =>   |- ph