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	adantrlr 401	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ((ps /\ ta) /\ ch)) -> th)
Theorem	adantrrl 402	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ (ta /\ ch))) -> th)
Theorem	adantrrr 403	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ (ch /\ ta))) -> th)
Theorem	ad2antrr 404	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- (((ph /\ ch) /\ th) -> ps)
Theorem	ad2antlr 405	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- (((ch /\ ph) /\ th) -> ps)
Theorem	ad2antrl 406	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- ((ch /\ (ph /\ th)) -> ps)
Theorem	ad2antll 407	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- ((ch /\ (th /\ ph)) -> ps)
Theorem	ad2ant2l 408	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ (ta /\ ps)) -> ch)
Theorem	ad2ant2r 409	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ (ps /\ ta)) -> ch)
Theorem	ad2ant2lr 410	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ (ps /\ ta)) -> ch)
Theorem	ad2ant2rl 411	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ (ta /\ ps)) -> ch)
Theorem	simpll 412	Simplification of a conjunction.
|- (((ph /\ ps) /\ ch) -> ph)
Theorem	simplr 413	Simplification of a conjunction.
|- (((ph /\ ps) /\ ch) -> ps)
Theorem	simprl 414	Simplification of a conjunction.
|- ((ph /\ (ps /\ ch)) -> ps)
Theorem	simprr 415	Simplification of a conjunction.
|- ((ph /\ (ps /\ ch)) -> ch)
Theorem	biimpa 416	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ps) -> ch)
Theorem	biimpar 417	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ch) -> ps)
Theorem	biimpac 418	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ps /\ ph) -> ch)
Theorem	biimparc 419	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ch /\ ph) -> ps)
Theorem	pm3.26bda 420	Deduction eliminating a conjunct.
|- (ph -> (ps <-> (ch /\ th)))   =>   |- ((ph /\ ps) -> ch)
Theorem	pm3.27bda 421	Deduction eliminating a conjunct.
|- (ph -> (ps <-> (ch /\ th)))   =>   |- ((ph /\ ps) -> th)
Theorem	jaob 422	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121.
|- (((ph \/ ch) -> ps) <-> ((ph -> ps) /\ (ch -> ps)))
Theorem	pm4.77 423	Theorem *4.77 of [WhiteheadRussell] p. 121.
|- (((ps -> ph) /\ (ch -> ph)) <-> ((ps \/ ch) -> ph))
Theorem	jaod 424	Deduction disjoining the antecedents of two implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ch))   =>   |- (ph -> ((ps \/ th) -> ch))
Theorem	jaoian 425	Inference disjoining the antecedents of two implications.
|- ((ph /\ ps) -> ch)   &   |- ((th /\ ps) -> ch)   =>   |- (((ph \/ th) /\ ps) -> ch)
Theorem	jaodan 426	Deduction disjoining the antecedents of two implications.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ th) -> ch)   =>   |- ((ph /\ (ps \/ th)) -> ch)
Theorem	jaao 427	Inference conjoining and disjoining the antecedents of two implications.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   =>   |- ((ph /\ th) -> ((ps \/ ta) -> ch))
Theorem	pm2.63 428	Theorem *2.63 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> ((-. ph \/ ps) -> ps))
Theorem	pm2.64 429	Theorem *2.64 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> ((ph \/ -. ps) -> ph))
Theorem	pm3.44 430	Theorem *3.44 of [WhiteheadRussell] p. 113.
|- (((ps -> ph) /\ (ch -> ph)) -> ((ps \/ ch) -> ph))
Theorem	pm4.43 431	Theorem *4.43 of [WhiteheadRussell] p. 119.
|- (ph <-> ((ph \/ ps) /\ (ph \/ -. ps)))
Theorem	anidm 432	Idempotent law for conjunction.
|- ((ph /\ ph) <-> ph)
Theorem	pm4.24 433	Theorem *4.24 of [WhiteheadRussell] p. 117.
|- (ph <-> (ph /\ ph))
Theorem	anidms 434	Inference from idempotent law for conjunction.
|- ((ph /\ ph) -> ps)   =>   |- (ph -> ps)
Theorem	ancom 435	Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118.
|- ((ph /\ ps) <-> (ps /\ ph))
Theorem	ancoms 436	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 437	Deduction commuting conjunction in antecedent.
|- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> ((ch /\ ps) -> th))
Theorem	pm3.22 438	Theorem *3.22 of [WhiteheadRussell] p. 111.
|- ((ph /\ ps) -> (ps /\ ph))
Theorem	anass 439	Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118.
|- (((ph /\ ps) /\ ch) <-> (ph /\ (ps /\ ch)))
Theorem	anasss 440	Associative law for conjunction applied to antecedent (eliminates syllogism).
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch)) -> th)
Theorem	anassrs 441	Associative law for conjunction applied to antecedent (eliminates syllogism).
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- (((ph /\ ps) /\ ch) -> th)
Theorem	imdistan 442	Distribution of implication with conjunction.
|- ((ph -> (ps -> ch)) <-> ((ph /\ ps) -> (ph /\ ch)))
Theorem	imdistani 443	Distribution of implication with conjunction.
|- (ph -> (ps -> ch))   =>   |- ((ph /\ ps) -> (ph /\ ch))
Theorem	imdistanri 444	Distribution of implication with conjunction.
|- (ph -> (ps -> ch))   =>   |- ((ps /\ ph) -> (ch /\ ph))
Theorem	imdistand 445	Distribution of implication with conjunction (deduction rule).
|- (ph -> (ps -> (ch -> th)))   =>   |- (ph -> ((ps /\ ch) -> (ps /\ th)))
Theorem	pm5.3 446	Theorem *5.3 of [WhiteheadRussell] p. 125.
|- (((ph /\ ps) -> ch) <-> ((ph /\ ps) -> (ph /\ ch)))
Theorem	pm5.61 447	Theorem *5.61 of [WhiteheadRussell] p. 125.
|- (((ph \/ ps) /\ -. ps) <-> (ph /\ -. ps))
Theorem	sylan 448	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ph)   =>   |- ((th /\ ps) -> ch)
Theorem	sylanb 449	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ph)   =>   |- ((th /\ ps) -> ch)
Theorem	sylanbr 450	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ph <-> th)   =>   |- ((th /\ ps) -> ch)
Theorem	sylan2 451	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ps)   =>   |- ((ph /\ th) -> ch)
Theorem	sylan2b 452	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ps)   =>   |- ((ph /\ th) -> ch)
Theorem	sylan2br 453	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ps <-> th)   =>   |- ((ph /\ th) -> ch)
Theorem	syl2an 454	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ph)   &   |- (ta -> ps)   =>   |- ((th /\ ta) -> ch)
Theorem	syl2anb 455	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ph)   &   |- (ta <-> ps)   =>   |- ((th /\ ta) -> ch)
Theorem	syl2anbr 456	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ph <-> th)   &   |- (ps <-> ta)   =>   |- ((th /\ ta) -> ch)
Theorem	syland 457	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ps))   =>   |- (ph -> ((ta /\ ch) -> th))
Theorem	sylan2d 458	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ch))   =>   |- (ph -> ((ps /\ ta) -> th))
Theorem	syl2and 459	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ps))   &   |- (ph -> (et -> ch))   =>   |- (ph -> ((ta /\ et) -> th))
Theorem	sylanl1 460	A syllogism inference.
|- ((