mmtheorems5 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 401-500 - Page 5 of 175

Type	Label	Description
Statement
Theorem	imp5d 401	An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
|- (ph -> (ps -> (ch -> (th -> (ta -> et)))))   =>   |- (((ph /\ ps) /\ ch) -> ((th /\ ta) -> et))
Theorem	ex 402	Exportation inference. (This theorem used to be labeled "exp" but was changed to "ex" so as not to conflict with the math token "exp", per the June 2006 Metamath spec change.) (The proof was shortened by Eric Schmidt, 22-Dec-2006.)
|- ((ph /\ ps) -> ch)   =>   |- (ph -> (ps -> ch))
Theorem	expcom 403	Exportation inference with commuted antecedents.
|- ((ph /\ ps) -> ch)   =>   |- (ps -> (ph -> ch))
Theorem	expimpd 404	Exportation followed by a deduction version of importation.
|- ((ph /\ ps) -> (ch -> th))   =>   |- (ph -> ((ps /\ ch) -> th))
Theorem	exp3a 405	Exportation deduction.
|- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ps -> (ch -> th)))
Theorem	expdimp 406	A deduction version of exportation, followed by importation.
|- (ph -> ((ps /\ ch) -> th))   =>   |- ((ph /\ ps) -> (ch -> th))
Theorem	exp31 407	An exportation inference.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (ph -> (ps -> (ch -> th)))
Theorem	exp32 408	An exportation inference.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- (ph -> (ps -> (ch -> th)))
Theorem	exp4a 409	An exportation inference.
|- (ph -> (ps -> ((ch /\ th) -> ta)))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp4b 410	An exportation inference.
|- ((ph /\ ps) -> ((ch /\ th) -> ta))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp4c 411	An exportation inference.
|- (ph -> (((ps /\ ch) /\ th) -> ta))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp4d 412	An exportation inference.
|- (ph -> ((ps /\ (ch /\ th)) -> ta))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp41 413	An exportation inference.
|- ((((ph /\ ps) /\ ch) /\ th) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp42 414	An exportation inference.
|- (((ph /\ (ps /\ ch)) /\ th) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp43 415	An exportation inference.
|- (((ph /\ ps) /\ (ch /\ th)) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp44 416	An exportation inference.
|- ((ph /\ ((ps /\ ch) /\ th)) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp45 417	An exportation inference.
|- ((ph /\ (ps /\ (ch /\ th))) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	expr 418	Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ps) -> (ch -> th))
Theorem	exp53 419	An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
|- ((((ph /\ ps) /\ (ch /\ th)) /\ ta) -> et)   =>   |- (ph -> (ps -> (ch -> (th -> (ta -> et)))))
Theorem	expl 420	Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (ph -> ((ps /\ ch) -> th))
Theorem	exbiri 421	Inference form of exbir 1285. This proof is exbiriVD 16678 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.)
|- ((ph /\ ps) -> (ch <-> th))   =>   |- (ph -> (ps -> (th -> ch)))
Theorem	impr 422	Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
|- ((ph /\ ps) -> (ch -> th))   =>   |- ((ph /\ (ps /\ ch)) -> th)
Theorem	impac 423	Importation with conjunction in consequent.
|- (ph -> (ps -> ch))   =>   |- ((ph /\ ps) -> (ch /\ ps))
Theorem	adantl 424	Inference adding a conjunct to the left of an antecedent.
|- (ph -> ps)   =>   |- ((ch /\ ph) -> ps)
Theorem	adantr 425	Inference adding a conjunct to the right of an antecedent.
|- (ph -> ps)   =>   |- ((ph /\ ch) -> ps)
Theorem	adantld 426	Deduction adding a conjunct to the left of an antecedent.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((th /\ ps) -> ch))
Theorem	adantrd 427	Deduction adding a conjunct to the right of an antecedent.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((ps /\ th) -> ch))
Theorem	adantll 428	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ ps) -> ch)
Theorem	adantlr 429	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ ps) -> ch)
Theorem	adantrl 430	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- ((ph /\ (th /\ ps)) -> ch)
Theorem	adantrr 431	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- ((ph /\ (ps /\ th)) -> ch)
Theorem	adantlll 432	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((((ta /\ ph) /\ ps) /\ ch) -> th)
Theorem	adantllr 433	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((((ph /\ ta) /\ ps) /\ ch) -> th)
Theorem	adantlrl 434	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ (ta /\ ps)) /\ ch) -> th)
Theorem	adantlrr 435	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ (ps /\ ta)) /\ ch) -> th)
Theorem	adantrll 436	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ((ta /\ ps) /\ ch)) -> th)
Theorem	adantrlr 437	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ((ps /\ ta) /\ ch)) -> th)
Theorem	adantrrl 438	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ (ta /\ ch))) -> th)
Theorem	adantrrr 439	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ (ch /\ ta))) -> th)
Theorem	ad2antrr 440	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- (((ph /\ ch) /\ th) -> ps)
Theorem	ad2antlr 441	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- (((ch /\ ph) /\ th) -> ps)
Theorem	ad2antrl 442	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- ((ch /\ (ph /\ th)) -> ps)
Theorem	ad2antll 443	Deduction adding conjuncts to antecedent.
|- (ph -> ps)   =>   |- ((ch /\ (th /\ ph)) -> ps)
Theorem	ad2ant2l 444	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ (ta /\ ps)) -> ch)
Theorem	ad2ant2r 445	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ (ps /\ ta)) -> ch)
Theorem	ad2ant2lr 446	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ (ps /\ ta)) -> ch)
Theorem	ad2ant2rl 447	Deduction adding two conjuncts to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ (ta /\ ps)) -> ch)
Theorem	simpll 448	Simplification of a conjunction.
|- (((ph /\ ps) /\ ch) -> ph)
Theorem	simplr 449	Simplification of a conjunction.
|- (((ph /\ ps) /\ ch) -> ps)
Theorem	simprl 450	Simplification of a conjunction.
|- ((ph /\ (ps /\ ch)) -> ps)
Theorem	simprr 451	Simplification of a conjunction.
|- ((ph /\ (ps /\ ch)) -> ch)
Theorem	simplll 452	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
|- ((((ph /\ ps) /\ ch) /\ th) -> ph)
Theorem	simpllr 453	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
|- ((((ph /\ ps) /\ ch) /\ th) -> ps)
Theorem	simplrl 454	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
|- (((ph /\ (ps /\ ch)) /\ th) -> ps)
Theorem	simplrr 455	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
|- (((ph /\ (ps /\ ch)) /\ th) -> ch)
Theorem	simprll 456	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
|- ((ph /\ ((ps /\ ch) /\ th)) -> ps)
Theorem	simprlr 457	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
|- ((ph /\ ((ps /\ ch) /\ th)) -> ch)
Theorem	simprrl 458	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
|- ((ph /\ (ps /\ (ch /\ th))) -> ch)
Theorem	simprrr 459	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
|- ((ph /\ (ps /\ (ch /\ th))) -> th)
Theorem	biimpa 460	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ps) -> ch)
Theorem	biimpar 461	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ch) -> ps)
Theorem	biimpac 462	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ps /\ ph) -> ch)
Theorem	biimparc 463	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ch /\ ph) -> ps)
Theorem	simprbda 464	Deduction eliminating a conjunct.
|- (ph -> (ps <-> (ch /\ th)))   =>   |- ((ph /\ ps) -> ch)
Theorem	simplbda 465	Deduction eliminating a conjunct.
|- (ph -> (ps <-> (ch /\ th)))   =>   |- ((ph /\ ps) -> th)
Theorem	simplbi2 466	Deduction eliminating a conjunct. Automatically derived from simplbi2VD 16670. (Contributed by Alan Sare, 31-Dec-2011.)
|- (ph <-> (ps /\ ch))   =>   |- (ps -> (ch -> ph))
Theorem	jaob 467	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121.
|- (((ph \/ ch) -> ps) <-> ((ph -> ps) /\ (ch -> ps)))
Theorem	pm4.77 468	Theorem *4.77 of [WhiteheadRussell] p. 121.
|- (((ps -> ph) /\ (ch -> ph)) <-> ((ps \/ ch) -> ph))
Theorem	jaod 469	Deduction disjoining the antecedents of two implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ch))   =>   |- (ph -> ((ps \/ th) -> ch))
Theorem	jaoian 470	Inference disjoining the antecedents of two implications.
|- ((ph /\ ps) -> ch)   &   |- ((th /\ ps) -> ch)   =>   |- (((ph \/ th) /\ ps) -> ch)
Theorem	jaodan 471	Deduction disjoining the antecedents of two implications.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ th) -> ch)   =>   |- ((ph /\ (ps \/ th)) -> ch)
Theorem	jaao 472	Inference conjoining and disjoining the antecedents of two implications.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   =>   |- ((ph /\ th) -> ((ps \/ ta) -> ch))
Theorem	jaoa 473	Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   =>   |- ((ph \/ th) -> ((ps /\ ta) -> ch))
Theorem	pm2.63 474	Theorem *2.63 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> ((-. ph \/ ps) -> ps))
Theorem	pm2.64 475	Theorem *2.64 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> ((ph \/ -. ps) -> ph))
Theorem	pm3.44 476	Theorem *3.44 of [WhiteheadRussell] p. 113.
|- (((ps -> ph) /\ (ch -> ph)) -> ((ps \/ ch) -> ph))
Theorem	pm4.43 477	Theorem *4.43 of [WhiteheadRussell] p. 119.
|- (ph <-> ((ph \/ ps) /\ (ph \/ -. ps)))
Theorem	anidm 478	Idempotent law for conjunction.
|- ((ph /\ ph) <-> ph)
Theorem	pm4.24 479	Theorem *4.24 of [WhiteheadRussell] p. 117.
|- (ph <-> (ph /\ ph))
Theorem	anidms 480	Inference from idempotent law for conjunction.
|- ((ph /\ ph) -> ps)   =>   |- (ph -> ps)
Theorem	anidmdbi 481	Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
|- ((ph -> (ps /\ ps)) <-> (ph -> ps))
Theorem	ancom 482	Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118.
|- ((ph /\ ps) <-> (ps /\ ph))
Theorem	ancomd 483	Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
|- (ph -> (ps /\ ch))   =>   |- (ph -> (ch /\ ps))
Theorem	ancoms 484	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 12) -type inference in a proof.
|- ((ph /\ ps) -> ch)   =>   |- ((ps /\ ph) -> ch)
Theorem	ancomsd 485	Deduction commuting conjunction in antecedent.
|- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> ((ch /\ ps) -> th))
Theorem	pm3.22 486	Theorem *3.22 of [WhiteheadRussell] p. 111.
|- ((ph /\ ps) -> (ps /\ ph))
Theorem	anass 487	Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118.
|- (((ph /\ ps) /\ ch) <-> (ph /\ (ps /\ ch)))
Theorem	anasss 488	Associative law for conjunction applied to antecedent (eliminates syllogism).
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch)) -> th)
Theorem	anassrs 489	Associative law for conjunction applied to antecedent (eliminates syllogism).
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- (((ph /\ ps) /\ ch) -> th)
Theorem	imdistan 490	Distribution of implication with conjunction.
|- ((ph -> (ps -> ch)) <-> ((ph /\ ps) -> (ph /\ ch)))
Theorem	imdistani 491	Distribution of implication with conjunction.
|- (ph -> (ps -> ch))   =>   |- ((ph /\ ps) -> (ph /\ ch))
Theorem	imdistanri 492	Distribution of implication with conjunction.
|- (ph -> (ps -> ch))   =>   |- ((ps /\ ph) -> (ch /\ ph))
Theorem	imdistand 493	Distribution of implication with conjunction (deduction rule).
|- (ph -> (ps -> (ch -> th)))   =>   |- (ph -> ((ps /\ ch) -> (ps /\ th)))
Theorem	pm5.3 494	Theorem *5.3 of [WhiteheadRussell] p. 125. (The proof was shortened by Andrew Salmon, 7-May-2011.)
|- (((ph /\ ps) -> ch) <-> ((ph /\ ps) -> (ph /\ ch)))
Theorem	pm5.3OLD 495	Theorem *5.3 of [WhiteheadRussell] p. 125.
|- (((ph /\ ps) -> ch) <-> ((ph /\ ps) -> (ph /\ ch)))
Theorem	pm5.61 496	Theorem *5.61 of [WhiteheadRussell] p. 125.
|- (((ph \/ ps) /\ -. ps) <-> (ph /\ -. ps))
Theorem	sylan 497	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ph)   =>   |- ((th /\ ps) -> ch)
Theorem	sylanb 498	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ph)   =>   |- ((th /\ ps) -> ch)
Theorem	sylanbr 499	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ph <-> th)   =>   |- ((th /\ ps) -> ch)
Theorem	sylan2 500	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ps)   =>   |- ((ph /\ th) -> ch)