Metamath Proof Explorer - mmtheorems5

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Statement List for Metamath Proof Explorer - 401-500 - Page 5 of 58

Type	Label	Description
Statement
Theorem	bicon4i 401	A contraposition inference.
|- (-. ph <-> -. ps)   =>   |- (ph <-> ps)
Theorem	bicon4d 402	A contraposition deduction.
|- (ph -> (-. ps <-> -. ch))   =>   |- (ph -> (ps <-> ch))
Theorem	bicon2 403	Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117.
|- ((ph <-> -. ps) <-> (ps <-> -. ph))
Theorem	bicon2d 404	A contraposition deduction.
|- (ph -> (ps <-> -. ch))   =>   |- (ph -> (ch <-> -. ps))
Theorem	bicon1d 405	A contraposition deduction.
|- (ph -> (-. ps <-> ch))   =>   |- (ph -> (-. ch <-> ps))
Theorem	bitrd 406	Deduction form of bitr 151.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   =>   |- (ph -> (ps <-> th))
Theorem	bitr2d 407	Deduction form of bitr2 152.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   =>   |- (ph -> (th <-> ps))
Theorem	bitr3d 408	Deduction form of bitr3 153.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ps <-> th))   =>   |- (ph -> (ch <-> th))
Theorem	bitr4d 409	Deduction form of bitr4 154.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ch))   =>   |- (ph -> (ps <-> th))
Theorem	syl5bb 410	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   =>   |- (ph -> (th <-> ch))
Theorem	syl5rbb 411	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   =>   |- (ph -> (ch <-> th))
Theorem	syl5bbr 412	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   =>   |- (ph -> (th <-> ch))
Theorem	syl5rbbr 413	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   =>   |- (ph -> (ch <-> th))
Theorem	syl6bb 414	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ch <-> th)   =>   |- (ph -> (ps <-> th))
Theorem	syl6rbb 415	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (ch <-> th)   =>   |- (ph -> (th <-> ps))
Theorem	syl6bbr 416	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ch)   =>   |- (ph -> (ps <-> th))
Theorem	syl6rbbr 417	A syllogism inference from two biconditionals.
|- (ph -> (ps <-> ch))   &   |- (th <-> ch)   =>   |- (ph -> (th <-> ps))
Theorem	sylan9bb 418	Nested syllogism inference conjoining dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta ))   =>   |- ((ph /\ th) -> (ps <-> ta ))
Theorem	sylan9bbr 419	Nested syllogism inference conjoining dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta ))   =>   |- ((th /\ ph) -> (ps <-> ta ))
Theorem	3imtr3d 420	More general version of 3imtr3 191. Useful for converting conditional definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ph -> (ps <-> th))   &   |- (ph -> (ch <-> ta ))   =>   |- (ph -> (th -> ta ))
Theorem	3imtr4d 421	More general version of 3imtr4 192. Useful for converting conditional definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ph -> (th <-> ps))   &   |- (ph -> (ta <-> ch))   =>   |- (ph -> (th -> ta ))
Theorem	3bitrd 422	Deduction from transitivity of biconditional.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch <-> th))   &   |- (ph -> (th <-> ta ))   =>   |- (ph -> (ps <-> ta ))
Theorem	3bitr3d 423	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	3bitr4d 424	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	3imtr3g 425	More general version of 3imtr3 191. Useful for converting definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (ps <-> th)   &   |- (ch <-> ta )   =>   |- (ph -> (th -> ta ))
Theorem	3imtr4g 426	More general version of 3imtr4 192. Useful for converting definitions in a formula.
|- (ph -> (ps -> ch))   &   |- (th <-> ps)   &   |- (ta <-> ch)   =>   |- (ph -> (th -> ta ))
Theorem	3bitr3g 427	More general version of 3bitr3 156. Useful for converting definitions in a formula.
|- (ph -> (ps <-> ch))   &   |- (ps <-> th)   &   |- (ch <-> ta )   =>   |- (ph -> (th <-> ta ))
Theorem	3bitr4g 428	More general version of 3bitr4 158. Useful for converting definitions in a formula.
|- (ph -> (ps <-> ch))   &   |- (th <-> ps)   &   |- (ta <-> ch)   =>   |- (ph -> (th <-> ta ))
Theorem	prth 429	Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it 'praeclarum theorema.'
|- (((ph -> ps) /\ (ch -> th)) -> ((ph /\ ch) -> (ps /\ th)))
Theorem	pm3.48 430	Theorem *3.48 of [WhiteheadRussell] p. 114.
|- (((ph -> ps) /\ (ch -> th)) -> ((ph \/ ch) -> (ps \/ th)))
Theorem	anim12d 431	Conjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta ))   =>   |- (ph -> ((ps /\ th) -> (ch /\ ta )))
Theorem	anim1d 432	Add a conjunct to right of antecedent and consequent in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((ps /\ th) -> (ch /\ th)))
Theorem	anim2d 433	Add a conjunct to left of antecedent and consequent in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((th /\ ps) -> (th /\ ch)))
Theorem	im2anan9 434	Deduction joining nested implications to form implication of conjunctions.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> et))   =>   |- ((ph /\ th) -> ((ps /\ ta ) -> (ch /\ et)))
Theorem	im2anan9r 435	Deduction joining nested implications to form implication of conjunctions.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> et))   =>   |- ((th /\ ph) -> ((ps /\ ta ) -> (ch /\ et)))
Theorem	orim12d 436	Disjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta ))   =>   |- (ph -> ((ps \/ th) -> (ch \/ ta )))
Theorem	orim1d 437	Disjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((ps \/ th) -> (ch \/ th)))
Theorem	orim2d 438	Disjoin antecedents and consequents in a deduction.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((th \/ ps) -> (th \/ ch)))
Theorem	pm2.85 439	Theorem *2.85 of [WhiteheadRussell] p. 108.
|- (((ph \/ ps) -> (ph \/ ch)) -> (ph \/ (ps -> ch)))
Theorem	pm3.2ni 440	Infer negated disjunction of negated premises.
|- -. ph   &   |- -. ps   =>   |- -. (ph \/ ps)
Theorem	oel 441	Elimination of redundant internal disjunct. Compare Theorem *4.45 of [WhiteheadRussell] p. 119.
|- (ph <-> ((ph \/ ps) /\ ph))
Theorem	pm5.74 442	Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126.
|- ((ph -> (ps <-> ch)) <-> ((ph -> ps) <-> (ph -> ch)))
Theorem	pm5.74i 443	Distribution of implication over biconditional (inference rule).
|- (ph -> (ps <-> ch))   =>   |- ((ph -> ps) <-> (ph -> ch))
Theorem	pm5.74d 444	Distribution of implication over biconditional (deduction rule).
|- (ph -> (ps -> (ch <-> th)))   =>   |- (ph -> ((ps -> ch) <-> (ps -> th)))
Theorem	pm5.74ri 445	Distribution of implication over biconditional (reverse inference rule).
|- ((ph -> ps) <-> (ph -> ch))   =>   |- (ph -> (ps <-> ch))
Theorem	pm5.74rd 446	Distribution of implication over biconditional (deduction rule).
|- (ph -> ((ps -> ch) <-> (ps -> th)))   =>   |- (ph -> (ps -> (ch <-> th)))
Theorem	mpbidi 447	A deduction from a biconditional, related to modus ponens.
|- (th -> (ph -> ps))   &   |- (ph -> (ps <-> ch))   =>   |- (th -> (ph -> ch))
Theorem	ibib 448	Implication in terms of implication and biconditional.
|- ((ph -> ps) <-> (ph -> (ph <-> ps)))
Theorem	ibi 449	Inference that converts a biconditional implied by one of its arguments, into an implication.
|- (ph -> (ph <-> ps))   =>   |- (ph -> ps)
Theorem	ibir 450	Inference that converts a biconditional implied by one of its arguments, into an implication.
|- (ph -> (ps <-> ph))   =>   |- (ph -> ps)
Theorem	ibd 451	Deduction that converts a biconditional implied by one of its arguments, into an implication.
|- (ph -> (ps -> (ps <-> ch)))   =>   |- (ph -> (ps -> ch))
Theorem	ordi 452	Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119.
|- ((ph \/ (ps /\ ch)) <-> ((ph \/ ps) /\ (ph \/ ch)))
Theorem	ordir 453	Distributive law for disjunction.
|- (((ph /\ ps) \/ ch) <-> ((ph \/ ch) /\ (ps \/ ch)))
Theorem	jcab 454	Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121.
|- ((ph -> (ps /\ ch)) <-> ((ph -> ps) /\ (ph -> ch)))
Theorem	jcad 455	Deduction conjoining the consequents of two implications.
|- (ph -> (ps -> ch))   &   |- (ph