Metamath Proof Explorer - mmtheorems4

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Statement List for Metamath Proof Explorer - 301-400 - Page 4 of 58

Type	Label	Description
Statement
Theorem	exp43 301	An exportation inference.
|- (((ph /\ ps) /\ (ch /\ th)) -> ta )   =>   |- (ph -> (ps -> (ch -> (th -> ta ))))
Theorem	exp44 302	An exportation inference.
|- ((ph /\ ((ps /\ ch) /\ th)) -> ta )   =>   |- (ph -> (ps -> (ch -> (th -> ta ))))
Theorem	exp45 303	An exportation inference.
|- ((ph /\ (ps /\ (ch /\ th))) -> ta )   =>   |- (ph -> (ps -> (ch -> (th -> ta ))))
Theorem	impac 304	Importation with conjunction in consequent.
|- (ph -> (ps -> ch))   =>   |- ((ph /\ ps) -> (ch /\ ps))
Theorem	adantl 305	Inference adding a conjunct to the left of an antecedent.
|- (ph -> ps)   =>   |- ((ch /\ ph) -> ps)
Theorem	adantr 306	Inference adding a conjunct to the right of an antecedent.
|- (ph -> ps)   =>   |- ((ph /\ ch) -> ps)
Theorem	adantld 307	Deduction adding a conjunct to the left of an antecedent.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((th /\ ps) -> ch))
Theorem	adantrd 308	Deduction adding a conjunct to the right of an antecedent.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((ps /\ th) -> ch))
Theorem	adantll 309	Deduction adding a conjunct to an antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ ps) -> ch)
Theorem	adantlr 310	Deduction adding a conjunct to an antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ ps) -> ch)
Theorem	adantrl 311	Deduction adding a conjunct to an antecedent.
|- ((ph /\ ps) -> ch)   =>   |- ((ph /\ (th /\ ps)) -> ch)
Theorem	adantrr 312	Deduction adding a conjunct to an antecedent.
|- ((ph /\ ps) -> ch)   =>   |- ((ph /\ (ps /\ th)) -> ch)
Theorem	adantlll 313	Deduction adding a conjunct to an antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((((ta /\ ph) /\ ps) /\ ch) -> th)
Theorem	adantlrl 314	Deduction adding a conjunct to an antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ (ta /\ ps)) /\ ch) -> th)
Theorem	adantlrr 315	Deduction adding a conjunct to an antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ (ps /\ ta )) /\ ch) -> th)
Theorem	adantrll 316	Deduction adding a conjunct to an antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ((ta /\ ps) /\ ch)) -> th)
Theorem	adantrlr 317	Deduction adding a conjunct to an antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ((ps /\ ta ) /\ ch)) -> th)
Theorem	adantrrl 318	Deduction adding a conjunct to an antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ (ta /\ ch))) -> th)
Theorem	adantrrr 319	Deduction adding a conjunct to an antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ (ch /\ ta ))) -> th)
Theorem	ad2antll 320	Deduction adding conjuncts to an antecedent.
|- (ph -> ps)   =>   |- (((ph /\ ch) /\ th) -> ps)
Theorem	ad2antlr 321	Deduction adding conjuncts to an antecedent.
|- (ph -> ps)   =>   |- (((ch /\ ph) /\ th) -> ps)
Theorem	ad2antrl 322	Deduction adding conjuncts to an antecedent.
|- (ph -> ps)   =>   |- ((ch /\ (ph /\ th)) -> ps)
Theorem	ad2antrr 323	Deduction adding conjuncts to an antecedent.
|- (ph -> ps)   =>   |- ((ch /\ (th /\ ph)) -> ps)
Theorem	biimpa 324	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ps) -> ch)
Theorem	biimpar 325	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ph /\ ch) -> ps)
Theorem	biimpac 326	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ps /\ ph) -> ch)
Theorem	biimparc 327	Inference from a logical equivalence.
|- (ph -> (ps <-> ch))   =>   |- ((ch /\ ph) -> ps)
Theorem	jaob 328	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121.
|- (((ph \/ ch) -> ps) <-> ((ph -> ps) /\ (ch -> ps)))
Theorem	jaod 329	Deduction disjoining the antecedents of two implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ch))   =>   |- (ph -> ((ps \/ th) -> ch))
Theorem	jaao 330	Inference conjoining and disjoining the antecedents of two implications.
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   =>   |- ((ph /\ th) -> ((ps \/ ta ) -> ch))
Theorem	anidm 331	Idempotent law for conjunction. Theorem *4.24 of [WhiteheadRussell] p. 117.
|- ((ph /\ ph) <-> ph)
Theorem	anidms 332	Inference from idempotent law for conjunction.
|- ((ph /\ ph) -> ps)   =>   |- (ph -> ps)
Theorem	ancom 333	Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118.
|- ((ph /\ ps) <-> (ps /\ ph))
Theorem	ancoms 334	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 335	Deduction commuting conjunction in antecedent.
|- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> ((ch /\ ps) -> th))
Theorem	anass 336	Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118.
|- (((ph /\ ps) /\ ch) <-> (ph /\ (ps /\ ch)))
Theorem	anasss 337	Associative law for conjunction applied to antecedent (eliminates syllogism).
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch)) -> th)
Theorem	anassrs 338	Associative law for conjunction applied to antecedent (eliminates syllogism).
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- (((ph /\ ps) /\ ch) -> th)
Theorem	imdistan 339	Distribution of implication with conjunction.
|- ((ph -> (ps -> ch)) <-> ((ph /\ ps) -> (ph /\ ch)))
Theorem	imdistani 340	Distribution of implication with conjunction.
|- (ph -> (ps -> ch))   =>   |- ((ph /\ ps) -> (ph /\ ch))
Theorem	imdistanri 341	Distribution of implication with conjunction.
|- (ph -> (ps -> ch))   =>   |- ((ps /\ ph) -> (ch /\ ph))
Theorem	imdistand 342	Distribution of implication with conjunction (deduction rule).
|- (ph -> (ps -> (ch -> th)))   =>   |- (ph -> ((ps /\ ch) -> (ps /\ th)))
Theorem	sylan 343	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ph)   =>   |- ((th /\ ps) -> ch)
Theorem	sylanb 344	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ph)   =>   |- ((th /\ ps) -> ch)
Theorem	sylanbr 345	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ph <-> th)   =>   |- ((th /\ ps) -> ch)
Theorem	sylan2 346	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ps)   =>   |- ((ph /\ th) -> ch)
Theorem	sylan2b 347	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ps)   =>   |- ((ph /\ th) -> ch)
Theorem	sylan2br 348	A syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ps <-> th)   =>   |- ((ph /\ th) -> ch)
Theorem	syl2an 349	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th -> ph)   &   |- (ta -> ps)   =>   |- ((th /\ ta ) -> ch)
Theorem	syl2anb 350	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (th <-> ph)   &   |- (ta <-> ps)   =>   |- ((th /\ ta ) -> ch)
Theorem	syl2anbr 351	A double syllogism inference.
|- ((ph /\ ps) -> ch)   &   |- (ph <-> th)   &   |- (ps <-> ta )   =>   |- ((th /\ ta ) -> ch)
Theorem	syland 352	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ps))   =>   |- (ph -> ((ta /\ ch) -> th))
Theorem	sylan2d 353	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ch))   =>   |- (ph -> ((ps /\ ta ) -> th))
Theorem	syl2and 354	A syllogism deduction.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ph -> (ta -> ps))   &   |- (ph -> (et -> ch))   =>   |- (ph -> ((ta /\ et) -> th))
Theorem	sylan12 355	A syllogism inference.
|- (((ph /\ ps) /\ ch) -> th)   &   |- (ta -> ps)   =>   |- (((ph /\ ta ) /\ ch) -> th)
Theorem	sylani 356	A syllogism inference.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ta -> ps)   =>   |- (ph -> ((ta /\ ch) -> th))
Theorem	sylan2i 357	A syllogism inference.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ta -> ch)   =>   |- (ph -> ((ps /\ ta ) -> th))
Theorem	syl2ani 358	A syllogism inference.
|- (ph -> ((ps /\ ch) -> th))   &   |- (ta -> ps)   &   |- (et -> ch)   =>   |- (ph -> ((ta /\ et) -> th))
Theorem	sylan9 359	Nested syllogism inference conjoining dissimilar antecedents.