mmtheorems5 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 401-500 - Page 5 of 108

Type	Label	Description
Statement
Theorem	adantlrr 401	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ (ps /\ ta)) /\ ch) -> th)
Theorem	adantrll 402	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ((ta /\ ps) /\ ch)) -> th)
Theorem	adantrlr 403	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ((ps /\ ta) /\ ch)) -> th)
Theorem	adantrrl 404	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ (ta /\ ch))) -> th)
Theorem	adantrrr 405	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ (ch /\ ta))) -> th)
Theorem	ad2antrr 406	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- (((ph /\ ch) /\ th) -> ps)
Theorem	ad2antlr 407	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- (((ch /\ ph) /\ th) -> ps)
Theorem	ad2antrl 408	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- ((ch /\ (ph /\ th)) -> ps)
Theorem	ad2antll 409	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- ((ch /\ (th /\ ph)) -> ps)
Theorem	ad2ant2l 410	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ (ta /\ ps)) -> ch)
Theorem	ad2ant2r 411	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ (ps /\ ta)) -> ch)
Theorem	ad2ant2lr 412	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ (ps /\ ta)) -> ch)
Theorem	ad2ant2rl 413	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ (ta /\ ps)) -> ch)
Theorem	simpll 414	Simplification of a conjunction.
|- (((ph /\ ps) /\ ch) -> ph)
Theorem	simplr 415	Simplification of a conjunction.
|- (((ph /\ ps) /\ ch) -> ps)
Theorem	simprl 416	Simplification of a conjunction.
|- ((ph /\ (ps /\ ch)) -> ps)
Theorem	simprr 417	Simplification of a conjunction.
|- ((ph /\ (ps /\ ch)) -> ch)
Theorem	biimpa 418	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ps) -> ch)
Theorem	biimpar 419	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ch) -> ps)
Theorem	biimpac 420	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ps /\ ph) -> ch)
Theorem	biimparc 421	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ch /\ ph) -> ps)
Theorem	pm3.26bda 422	Deduction eliminating a conjunct.
|- (ph -> (ps <-> (ch /\ th)))   =>   |- ((ph /\ ps) -> ch)
Theorem	pm3.27bda 423	Deduction eliminating a conjunct.
|- (ph -> (ps <-> (ch /\ th)))   =>   |- ((ph /\ ps) -> th)
Theorem	jaob 424	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121.
|- (((ph \/ ch) -> ps) <-> ((ph -> ps) /\ (ch -> ps)))
Theorem	pm4.77 425	Theorem *4.77 of [WhiteheadRussell] p. 121.
|- (((ps -> ph) /\ (ch -> ph)) <-> ((ps \/ ch) -> ph))
Theorem	jaod 426	Deduction disjoining the antecedents of two implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ch))   =>   |- (ph -> ((ps \/ th) -> ch))
Theorem	jaoian 427	Inference disjoining the antecedents of two implications.
|- ((ph /\ ps) -> ch)   &   |- ((th /\ ps) -> ch)   =>   |- (((ph \/ th) /\ ps) -> ch)
Theorem	jaodan 428	Deduction disjoining the antecedents of two implications.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ th) -> ch)   =>   |- ((ph /\ (ps \/ th)) -> ch)
Theorem	jaao 429	Inference conjoining and disjoining the antecedents of two implications.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   =>   |- ((ph /\ th) -> ((ps \/ ta) -> ch))
Theorem	pm2.63 430	Theorem *2.63 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> ((-. ph \/ ps) -> ps))
Theorem	pm2.64 431	Theorem *2.64 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> ((ph \/ -. ps) -> ph))
Theorem	pm3.44 432	Theorem *3.44 of [WhiteheadRussell] p. 113.
|- (((ps -> ph) /\ (ch -> ph)) -> ((ps \/ ch) -> ph))
Theorem	pm4.43 433	Theorem *4.43 of [WhiteheadRussell] p. 119.
|- (ph <-> ((ph \/ ps) /\ (ph \/ -. ps)))
Theorem	anidm 434	Idempotent law for conjunction.
|- ((ph /\ ph) <-> ph)
Theorem	pm4.24 435	Theorem *4.24 of [WhiteheadRussell] p. 117.
|- (ph <-> (ph /\ ph))
Theorem	anidms 436	Inference from idempotent law for conjunction.
|- ((ph /\ ph) -> ps)   =>   |- (ph -> ps)
Theorem	ancom 437	Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118.
|- ((ph /\ ps) <-> (ps /\ ph))
Theorem	ancoms 438	Inference commuting conjunction in antecedent. Notational convention: We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 10) -type inference in a proof.
|- ((ph /\ ps) -> ch)   =>   |- ((ps /\ ph) -> ch)
Theorem	ancomsd 439	Deduction commuting conjunction in antecedent.
|- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> ((ch /\ ps) -> th))
Theorem	pm3.22 440	Theorem *3.22 of [WhiteheadRussell] p. 111.
|- ((ph /\ ps) -> (ps /\ ph))
Theorem	anass 441	Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118.
|- (((ph /\ ps) /\ ch) <-> (ph /\ (ps /\ ch)))
Theorem	anasss 442	Associative law for conjunction applied to antecedent (eliminates syllogism).
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch)) -> th)
Theorem	anassrs 443	Associative law for conjunction applied to antecedent (eliminates syllogism).
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- (((ph /\ ps) /\ ch) -> th)
Theorem	imdistan 444	Distribution of implication with conjunction.
|- ((ph -> (ps -> ch)) <-> ((ph /\ ps) -> (ph /\ ch)))
Theorem	imdistani 445	Distribution of implication with conjunction.
|- (ph -> (ps -> ch))   =>   |- ((ph /\ ps) -> (ph /\ ch))
Theorem	imdistanri 446	Distribution of implication with conjunction.
|- (ph -> (ps -> ch))   =>   |- ((ps /\ ph) -> (ch /\ ph))
Theorem	imdistand 447	Distribution of implication with conjunction (deduction rule).
|- (ph -> (ps -> (ch -> th)))   =>   |- (ph -> ((ps /\ ch) -> (ps /\ th)))
Theorem	pm5.3 448	Theorem *5.3 of [WhiteheadRussell] p. 125.
|- (((ph /\ ps) -> ch) <-> ((ph /\ ps) -> (ph /\ ch)))
Theorem	pm5.61 449	Theorem *5.61 of [WhiteheadRussell] p. 125.
|- (((ph \/ ps) /\ -. ps) <-> (ph /\ -. ps))
Theorem	sylan 450	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ph)   =>   |- ((th /\ ps) -> ch)
Theorem	sylanb 451	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ph)   =>   |- ((th /\ ps) -> ch)
Theorem	sylanbr 452	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ph <-> th)   =>   |- ((th /\ ps) -> ch)
Theorem	sylan2 453	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ps)   =>   |- ((ph /\ th) -> ch)
Theorem	sylan2b 454	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ps)   =>   |- ((ph /\ th) -> ch)
Theorem	sylan2br 455	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ps <-> th)   =>   |- ((ph /\ th) -> ch)
Theorem	syl2an 456	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ph)   &   |- (ta -> ps)   =>   |- ((th /\ ta) -> ch)
Theorem	syl2anb 457	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ph)   &   |- (ta <-> ps)   =>   |- ((th /\ ta) -> ch)
Theorem	syl2anbr 458	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ph <-> th)   &   |- (ps <-> ta)   =>   |- ((th /\ ta) -> ch)
Theorem	syland 459	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ps))   =>   |- (ph -> ((ta /\ ch) -> th))
Theorem	sylan2d 460	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ch))   =>   |- (ph -> ((ps /\