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.
⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → τ)    ⇒   ⊢ (φ → (ψ → (χ → (θ → τ))))
Theorem	exp44 302	An exportation inference.
⊢ ((φ ∧ ((ψ ∧ χ) ∧ θ)) → τ)    ⇒   ⊢ (φ → (ψ → (χ → (θ → τ))))
Theorem	exp45 303	An exportation inference.
⊢ ((φ ∧ (ψ ∧ (χ ∧ θ))) → τ)    ⇒   ⊢ (φ → (ψ → (χ → (θ → τ))))
Theorem	impac 304	Importation with conjunction in consequent.
⊢ (φ → (ψ → χ))    ⇒   ⊢ ((φ ∧ ψ) → (χ ∧ ψ))
Theorem	adantl 305	Inference adding a conjunct to the left of an antecedent.
⊢ (φ → ψ)    ⇒   ⊢ ((χ ∧ φ) → ψ)
Theorem	adantr 306	Inference adding a conjunct to the right of an antecedent.
⊢ (φ → ψ)    ⇒   ⊢ ((φ ∧ χ) → ψ)
Theorem	adantld 307	Deduction adding a conjunct to the left of an antecedent.
⊢ (φ → (ψ → χ))    ⇒   ⊢ (φ → ((θ ∧ ψ) → χ))
Theorem	adantrd 308	Deduction adding a conjunct to the right of an antecedent.
⊢ (φ → (ψ → χ))    ⇒   ⊢ (φ → ((ψ ∧ θ) → χ))
Theorem	adantll 309	Deduction adding a conjunct to an antecedent.
⊢ ((φ ∧ ψ) → χ)    ⇒   ⊢ (((θ ∧ φ) ∧ ψ) → χ)
Theorem	adantlr 310	Deduction adding a conjunct to an antecedent.
⊢ ((φ ∧ ψ) → χ)    ⇒   ⊢ (((φ ∧ θ) ∧ ψ) → χ)
Theorem	adantrl 311	Deduction adding a conjunct to an antecedent.
⊢ ((φ ∧ ψ) → χ)    ⇒   ⊢ ((φ ∧ (θ ∧ ψ)) → χ)
Theorem	adantrr 312	Deduction adding a conjunct to an antecedent.
⊢ ((φ ∧ ψ) → χ)    ⇒   ⊢ ((φ ∧ (ψ ∧ θ)) → χ)
Theorem	adantlll 313	Deduction adding a conjunct to an antecedent.
⊢ (((φ ∧ ψ) ∧ χ) → θ)    ⇒   ⊢ ((((τ ∧ φ) ∧ ψ) ∧ χ) → θ)
Theorem	adantlrl 314	Deduction adding a conjunct to an antecedent.
⊢ (((φ ∧ ψ) ∧ χ) → θ)    ⇒   ⊢ (((φ ∧ (τ ∧ ψ)) ∧ χ) → θ)
Theorem	adantlrr 315	Deduction adding a conjunct to an antecedent.
⊢ (((φ ∧ ψ) ∧ χ) → θ)    ⇒   ⊢ (((φ ∧ (ψ ∧ τ)) ∧ χ) → θ)
Theorem	adantrll 316	Deduction adding a conjunct to an antecedent.
⊢ ((φ ∧ (ψ ∧ χ)) → θ)    ⇒   ⊢ ((φ ∧ ((τ ∧ ψ) ∧ χ)) → θ)
Theorem	adantrlr 317	Deduction adding a conjunct to an antecedent.
⊢ ((φ ∧ (ψ ∧ χ)) → θ)    ⇒   ⊢ ((φ ∧ ((ψ ∧ τ) ∧ χ)) → θ)
Theorem	adantrrl 318	Deduction adding a conjunct to an antecedent.
⊢ ((φ ∧ (ψ ∧ χ)) → θ)    ⇒   ⊢ ((φ ∧ (ψ ∧ (τ ∧ χ))) → θ)
Theorem	adantrrr 319	Deduction adding a conjunct to an antecedent.
⊢ ((φ ∧ (ψ ∧ χ)) → θ)    ⇒   ⊢ ((φ ∧ (ψ ∧ (χ ∧ τ))) → θ)
Theorem	ad2antll 320	Deduction adding conjuncts to an antecedent.
⊢ (φ → ψ)    ⇒   ⊢ (((φ ∧ χ) ∧ θ) → ψ)
Theorem	ad2antlr 321	Deduction adding conjuncts to an antecedent.
⊢ (φ → ψ)    ⇒   ⊢ (((χ ∧ φ) ∧ θ) → ψ)
Theorem	ad2antrl 322	Deduction adding conjuncts to an antecedent.
⊢ (φ → ψ)    ⇒   ⊢ ((χ ∧ (φ ∧ θ)) → ψ)
Theorem	ad2antrr 323	Deduction adding conjuncts to an antecedent.
⊢ (φ → ψ)    ⇒   ⊢ ((χ ∧ (θ ∧ φ)) → ψ)
Theorem	biimpa 324	Inference from a logical equivalence.
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	biimpar 325	Inference from a logical equivalence.
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ ((φ ∧ χ) → ψ)
Theorem	biimpac 326	Inference from a logical equivalence.
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ ((ψ ∧ φ) → χ)
Theorem	biimparc 327	Inference from a logical equivalence.
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ ((χ ∧ φ) → ψ)
Theorem	jaob 328	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121.
⊢ (((φ ∨ χ) → ψ) ↔ ((φ → ψ) ∧ (χ → ψ)))
Theorem	jaod 329	Deduction disjoining the antecedents of two implications.
⊢ (φ → (ψ → χ))    &   ⊢ (φ → (θ → χ))    ⇒   ⊢ (φ → ((ψ ∨ θ) → χ))
Theorem	jaao 330	Inference conjoining and disjoining the antecedents of two implications.
⊢ (φ → (ψ → χ))    &   ⊢ (θ → (τ → χ))    ⇒   ⊢ ((φ ∧ θ) → ((ψ ∨ τ) → χ))
Theorem	anidm 331	Idempotent law for conjunction. Theorem *4.24 of [WhiteheadRussell] p. 117.
⊢ ((φ ∧ φ) ↔ φ)
Theorem	anidms 332	Inference from idempotent law for conjunction.
⊢ ((φ ∧ φ) → ψ)    ⇒   ⊢ (φ → ψ)
Theorem	ancom 333	Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118.
⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
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.
⊢ ((φ ∧ ψ) → χ)    ⇒   ⊢ ((ψ ∧ φ) → χ)
Theorem	ancomsd 335	Deduction commuting conjunction in antecedent.
⊢ (φ → ((ψ ∧ χ) → θ))    ⇒   ⊢ (φ → ((χ ∧ ψ) → θ))
Theorem	anass 336	Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118.
⊢ (((φ ∧ ψ) ∧ χ) ↔ (φ ∧ (ψ ∧ χ)))
Theorem	anasss 337	Associative law for conjunction applied to antecedent (eliminates syllogism).
⊢ (((φ ∧ ψ) ∧ χ) → θ)    ⇒   ⊢ ((φ ∧ (ψ ∧ χ)) → θ)
Theorem	anassrs 338	Associative law for conjunction applied to antecedent (eliminates syllogism).
⊢ ((φ ∧ (ψ ∧ χ)) → θ)    ⇒   ⊢ (((φ ∧ ψ) ∧ χ) → θ)
Theorem	imdistan 339	Distribution of implication with conjunction.
⊢ ((φ → (ψ → χ)) ↔ ((φ ∧ ψ) → (φ ∧ χ)))
Theorem	imdistani 340	Distribution of implication with conjunction.
⊢ (φ → (ψ → χ))    ⇒   ⊢ ((φ ∧ ψ) → (φ ∧ χ))
Theorem	imdistanri 341	Distribution of implication with conjunction.
⊢ (φ → (ψ → χ))    ⇒   ⊢ ((ψ ∧ φ) → (χ ∧ φ))
Theorem	imdistand 342	Distribution of implication with conjunction (deduction rule).
⊢ (φ → (ψ → (χ → θ)))    ⇒   ⊢ (φ → ((ψ ∧ χ) → (ψ ∧ θ)))
Theorem	sylan 343	A syllogism inference.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (θ → φ)    ⇒   ⊢ ((θ ∧ ψ) → χ)
Theorem	sylanb 344	A syllogism inference.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (θ ↔ φ)    ⇒   ⊢ ((θ ∧ ψ) → χ)
Theorem	sylanbr 345	A syllogism inference.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (φ ↔ θ)    ⇒   ⊢ ((θ ∧ ψ) → χ)
Theorem	sylan2 346	A syllogism inference.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (θ → ψ)    ⇒   ⊢ ((φ ∧ θ) → χ)
Theorem	sylan2b 347	A syllogism inference.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (θ ↔ ψ)    ⇒   ⊢ ((φ ∧ θ) → χ)
Theorem	sylan2br 348	A syllogism inference.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (ψ ↔ θ)    ⇒   ⊢ ((φ ∧ θ) → χ)
Theorem	syl2an 349	A double syllogism inference.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (θ → φ)    &   ⊢ (τ → ψ)    ⇒   ⊢ ((θ ∧ τ) → χ)
Theorem	syl2anb 350	A double syllogism inference.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (θ ↔ φ)    &   ⊢ (τ ↔ ψ)    ⇒   ⊢ ((θ ∧ τ) → χ)
Theorem	syl2anbr 351	A double syllogism inference.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (φ ↔ θ)    &   ⊢ (ψ ↔ τ)    ⇒   ⊢ ((θ ∧ τ) → χ)
Theorem	syland 352	A syllogism deduction.
⊢ (φ → ((ψ ∧ χ) → θ))    &   ⊢ (φ → (τ → ψ))    ⇒   ⊢ (φ → ((τ ∧ χ) → θ))
Theorem	sylan2d 353	A syllogism deduction.
⊢ (φ → ((ψ ∧ χ) → θ))    &   ⊢ (φ → (τ → χ))    ⇒   ⊢ (φ → ((ψ ∧ τ) → θ))
Theorem	syl2and 354	A syllogism deduction.
⊢ (φ → ((ψ ∧ χ) → θ))    &   ⊢ (φ → (τ → ψ))    &   ⊢ (φ → (η → χ))    ⇒   ⊢ (φ → ((τ ∧ η) → θ))
Theorem	sylan12 355	A syllogism inference.
⊢ (((φ ∧ ψ) ∧ χ) → θ)    &   ⊢ (τ → ψ)    ⇒   ⊢ (((φ ∧ τ) ∧ χ) → θ)
Theorem	sylani 356	A syllogism inference.
⊢ (φ → ((ψ ∧ χ) → θ))    &   ⊢ (τ → ψ)    ⇒   ⊢ (φ → ((τ ∧ χ) → θ))
Theorem	sylan2i 357	A syllogism inference.
⊢ (φ → ((ψ ∧ χ) → θ))    &   ⊢ (τ → χ)    ⇒   ⊢ (φ → ((ψ ∧ τ) → θ))
Theorem	syl2ani 358	A syllogism inference.
⊢ (φ → ((ψ ∧ χ) → θ))    &   ⊢ (τ → ψ)    &   ⊢ (η → χ)    ⇒   ⊢ (φ → ((τ ∧ η) → θ))
Theorem	sylan9 359	Nested syllogism inference conjoining dissimilar antecedents.
⊢ (φ → (ψ → χ))    &   ⊢ (θ → (χ → τ))    ⇒   ⊢ ((φ ∧ θ) → (ψ → τ))
Theorem	sylan9r 360	Nested syllogism inference conjoining dissimilar antecedents.
⊢ (φ → (ψ → χ))    &   ⊢ (θ → (χ → τ))    ⇒   ⊢ ((θ ∧ φ) → (ψ → τ))
Theorem	sylanc 361	A syllogism inference combined with contraction.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (θ → φ)    &   ⊢ (θ → ψ)    ⇒   ⊢ (θ → χ)
Theorem	sylancb 362	A syllogism inference combined with contraction.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (θ ↔ φ)    &   ⊢ (θ ↔ ψ)    ⇒   ⊢ (θ → χ)
Theorem	sylancbr 363	A syllogism inference combined with contraction.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ (φ ↔ θ)    &   ⊢ (ψ ↔ θ)    ⇒   ⊢ (θ → χ)
Theorem	pm2.61an1 364	Elimination of an antecedent.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ ((¬ φ ∧ ψ) → χ)    ⇒   ⊢ (ψ → χ)
Theorem	pm2.61an2 365	Elimination of an antecedent.
⊢ ((φ ∧ ψ) → χ)    &   ⊢ ((φ ∧ ¬ ψ) → χ)    ⇒   ⊢ (φ → χ)
Theorem	abai 366	Introduce one conjunct as an antecedent to the another.
⊢ ((φ ∧ ψ) ↔ (φ ∧ (φ → ψ)))
Theorem	anbi2i 367	Introduce a left conjunct to both sides of a logical equivalence.
⊢ (φ ↔ ψ)    ⇒   ⊢ ((χ ∧ φ) ↔ (χ ∧ ψ))
Theorem	anbi1i 368	Introduce a right conjunct to both sides of a logical equivalence.
⊢ (φ ↔ ψ)    ⇒   ⊢ ((φ ∧ χ) ↔ (ψ ∧ χ))
Theorem	anbi12i 369	Conjoin both sides of two equivalences.
⊢ (φ ↔ ψ)    &   ⊢ (χ ↔ θ)    ⇒   ⊢ ((φ ∧ χ) ↔ (ψ ∧ θ))
Theorem	an12 370	A rearrangement of conjuncts.
⊢ ((φ ∧ (ψ ∧ χ)) ↔ (ψ ∧ (φ ∧ χ)))
Theorem	an23 371	A rearrangement of conjuncts.
⊢ (((φ ∧ ψ) ∧ χ) ↔ ((φ ∧ χ) ∧ ψ))
Theorem	an1s 372	Deduction rearranging conjuncts.
⊢ ((φ ∧ (ψ ∧ χ)) → θ)    ⇒   ⊢ ((ψ ∧ (φ ∧ χ)) → θ)
Theorem	an1rs 373	Deduction rearranging conjuncts.
⊢ (((φ ∧ ψ) ∧ χ) → θ)    ⇒   ⊢ (((φ ∧ χ) ∧ ψ) → θ)
Theorem	anabs1 374	Absorption into embedded conjunct.
⊢ (((φ ∧ ψ) ∧ φ) ↔ (φ ∧ ψ))
Theorem	anabs5 375	Absorption into embedded conjunct.
⊢ ((φ ∧ (φ ∧ ψ)) ↔ (φ ∧ ψ))
Theorem	anabs7 376	Absorption into embedded conjunct.
⊢ ((ψ ∧ (φ ∧ ψ)) ↔ (φ ∧ ψ))
Theorem	anabsi5 377	Absorption of antecedent into conjunction.
⊢ (φ → ((φ ∧ ψ) → χ))    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	anabsi6 378	Absorption of antecedent into conjunction.
⊢ (φ → ((ψ ∧ φ) → χ))    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	anabsi7 379	Absorption of antecedent into conjunction.
⊢ (ψ → ((φ ∧ ψ) → χ))    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	anabsi8 380	Absorption of antecedent into conjunction.
⊢ (ψ → ((ψ ∧ φ) → χ))    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	anabss1 381	Absorption of antecedent into conjunction.
⊢ (((φ ∧ ψ) ∧ φ) → χ)    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	anabss3 382	Absorption of antecedent into conjunction.
⊢ (((φ ∧ ψ) ∧ ψ) → χ)    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	anabss4 383	Absorption of antecedent into conjunction.
⊢ (((ψ ∧ φ) ∧ ψ) → χ)    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	anabss5 384	Absorption of antecedent into conjunction.
⊢ ((φ ∧ (φ ∧ ψ)) → χ)    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	anabss7 385	Absorption of antecedent into conjunction.
⊢ ((ψ ∧ (φ ∧ ψ)) → χ)    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	anabsan 386	Absorption of antecedent with conjunction.
⊢ (((φ ∧ φ) ∧ ψ) → χ)    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	anabsan2 387	Absorption of antecedent with conjunction.
⊢ ((φ ∧ (ψ ∧ ψ)) → χ)    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	an4 388	Rearrangement of 4 conjuncts.
⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ((φ ∧ χ) ∧ (ψ ∧ θ)))
Theorem	an42 389	Rearrangement of 4 conjuncts.
⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ((φ ∧ χ) ∧ (θ ∧ ψ)))
Theorem	an4s 390	Inference rearranging 4 conjuncts in antecedent.
⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → τ)    ⇒   ⊢ (((φ ∧ χ) ∧ (ψ ∧ θ)) → τ)
Theorem	an42s 391	Inference rearranging 4 conjuncts in antecedent.
⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → τ)    ⇒   ⊢ (((φ ∧ χ) ∧ (θ ∧ ψ)) → τ)
Theorem	anandi 392	Distribution of conjunction over conjunction.
⊢ ((φ ∧ (ψ ∧ χ)) ↔ ((φ ∧ ψ) ∧ (φ ∧ χ)))
Theorem	anandir 393	Distribution of conjunction over conjunction.
⊢ (((φ ∧ ψ) ∧ χ) ↔ ((φ ∧ χ) ∧ (ψ ∧ χ)))
Theorem	anandis 394	Inference that undistributes conjunction in the antecedent.
⊢ (((φ ∧ ψ) ∧ (φ ∧ χ)) → τ)    ⇒   ⊢ ((φ ∧ (ψ ∧ χ)) → τ)
Theorem	anandirs 395	Inference that undistributes conjunction in the antecedent.
⊢ (((φ ∧ χ) ∧ (ψ ∧ χ)) → τ)    ⇒   ⊢ (((φ ∧ ψ) ∧ χ) → τ)
Theorem	bi 396	A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49.
⊢ ((φ ↔ ψ) ↔ ((φ → ψ) ∧ (ψ → φ)))
Theorem	impbid 397	Deduce an equivalence from two implications.
⊢ (φ → (ψ → χ))    &   ⊢ (φ → (χ → ψ))    ⇒   ⊢ (φ → (ψ ↔ χ))
Theorem	bicom 398	Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117.
⊢ ((φ ↔ ψ) ↔ (ψ ↔ φ))
Theorem	bicomd 399	Commute two sides of a biconditional in a deduction.
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (χ ↔ ψ))
Theorem	pm4.11 400	Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117.
⊢ ((φ ↔ ψ) ↔ (¬ φ ↔ ¬ ψ))