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Theorem List for Intuitionistic Logic Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmpan2 401 An inference based on modus ponens. (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
ψ    &   ((φ ψ) → χ)       (φχ)

Theoremmp2an 402 An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
φ    &   ψ    &   ((φ ψ) → χ)       χ

Theoremmp4an 403 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2011.)
φ    &   ψ    &   χ    &   θ    &   (((φ ψ) (χ θ)) → τ)       τ

Theoremmpan2d 404 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(φχ)    &   (φ → ((ψ χ) → θ))       (φ → (ψθ))

Theoremmpand 405 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
(φψ)    &   (φ → ((ψ χ) → θ))       (φ → (χθ))

Theoremmpani 406 An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
ψ    &   (φ → ((ψ χ) → θ))       (φ → (χθ))

Theoremmpan2i 407 An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
χ    &   (φ → ((ψ χ) → θ))       (φ → (ψθ))

Theoremmp2ani 408 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
ψ    &   χ    &   (φ → ((ψ χ) → θ))       (φθ)

Theoremmp2and 409 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(φψ)    &   (φχ)    &   (φ → ((ψ χ) → θ))       (φθ)

Theoremmpanl1 410 An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
φ    &   (((φ ψ) χ) → θ)       ((ψ χ) → θ)

Theoremmpanl2 411 An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
ψ    &   (((φ ψ) χ) → θ)       ((φ χ) → θ)

Theoremmpanl12 412 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
φ    &   ψ    &   (((φ ψ) χ) → θ)       (χθ)

Theoremmpanr1 413 An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
ψ    &   ((φ (ψ χ)) → θ)       ((φ χ) → θ)

Theoremmpanr2 414 An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
χ    &   ((φ (ψ χ)) → θ)       ((φ ψ) → θ)

Theoremmpanr12 415 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
ψ    &   χ    &   ((φ (ψ χ)) → θ)       (φθ)

Theoremmpanlr1 416 An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
ψ    &   (((φ (ψ χ)) θ) → τ)       (((φ χ) θ) → τ)

Theorempm5.74da 417 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)
((φ ψ) → (χθ))       (φ → ((ψχ) ↔ (ψθ)))

Theoremimdistan 418 Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
((φ → (ψχ)) ↔ ((φ ψ) → (φ χ)))

Theoremimdistani 419 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
(φ → (ψχ))       ((φ ψ) → (φ χ))

Theoremimdistanri 420 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
(φ → (ψχ))       ((ψ φ) → (χ φ))

Theoremimdistand 421 Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.)
(φ → (ψ → (χθ)))       (φ → ((ψ χ) → (ψ θ)))

Theoremimdistanda 422 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)
((φ ψ) → (χθ))       (φ → ((ψ χ) → (ψ θ)))

Theorempm5.32d 423 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)
(φ → (ψ → (χθ)))       (φ → ((ψ χ) ↔ (ψ θ)))

Theorempm5.32rd 424 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.)
(φ → (ψ → (χθ)))       (φ → ((χ ψ) ↔ (θ ψ)))

Theorempm5.32da 425 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.)
((φ ψ) → (χθ))       (φ → ((ψ χ) ↔ (ψ θ)))

Theorempm5.32 426 Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by NM, 31-Jan-2015.)
((φ → (ψχ)) ↔ ((φ ψ) ↔ (φ χ)))

Theorempm5.32i 427 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
(φ → (ψχ))       ((φ ψ) ↔ (φ χ))

Theorempm5.32ri 428 Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)
(φ → (ψχ))       ((ψ φ) ↔ (χ φ))

(φψ)    &   (ψ → (φχ))       (φ ↔ (ψ χ))

Theoremanbi2i 430 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(φψ)       ((χ φ) ↔ (χ ψ))

Theoremanbi1i 431 Introduce a right conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(φψ)       ((φ χ) ↔ (ψ χ))

Theoremanbi2ci 432 Variant of anbi2i 430 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(φψ)       ((φ χ) ↔ (χ ψ))

Theoremanbi12i 433 Conjoin both sides of two equivalences. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χθ)       ((φ χ) ↔ (ψ θ))

Theoremanbi12ci 434 Variant of anbi12i 433 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φψ)    &   (χθ)       ((φ χ) ↔ (θ ψ))

Theoremsylan9bb 435 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(φ → (ψχ))    &   (θ → (χτ))       ((φ θ) → (ψτ))

Theoremsylan9bbr 436 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(φ → (ψχ))    &   (θ → (χτ))       ((θ φ) → (ψτ))

Theoremanbi2d 437 Deduction adding a left conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(φ → (ψχ))       (φ → ((θ ψ) ↔ (θ χ)))

Theoremanbi1d 438 Deduction adding a right conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(φ → (ψχ))       (φ → ((ψ θ) ↔ (χ θ)))

Theoremanbi1 439 Introduce a right conjunct to both sides of a logical equivalence. Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
((φψ) → ((φ χ) ↔ (ψ χ)))

Theoremanbi2 440 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
((φψ) → ((χ φ) ↔ (χ ψ)))

Theorembitr 441 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(((φψ) (ψχ)) → (φχ))

Theoremanbi12d 442 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (φ → (θτ))       (φ → ((ψ θ) ↔ (χ τ)))

Theoremmpan10 443 Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)
((((φψ) χ) φ) → (ψ χ))

Theorempm5.3 444 Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(((φ ψ) → χ) ↔ ((φ ψ) → (φ χ)))

Theoremadantll 445 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((φ ψ) → χ)       (((θ φ) ψ) → χ)

Theoremadantlr 446 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((φ ψ) → χ)       (((φ θ) ψ) → χ)

Theoremadantrl 447 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((φ ψ) → χ)       ((φ (θ ψ)) → χ)

Theoremadantrr 448 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((φ ψ) → χ)       ((φ (ψ θ)) → χ)

Theoremadantlll 449 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
(((φ ψ) χ) → θ)       ((((τ φ) ψ) χ) → θ)

Theoremadantllr 450 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
(((φ ψ) χ) → θ)       ((((φ τ) ψ) χ) → θ)

Theoremadantlrl 451 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
(((φ ψ) χ) → θ)       (((φ (τ ψ)) χ) → θ)

Theoremadantlrr 452 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
(((φ ψ) χ) → θ)       (((φ (ψ τ)) χ) → θ)

Theoremadantrll 453 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
((φ (ψ χ)) → θ)       ((φ ((τ ψ) χ)) → θ)

Theoremadantrlr 454 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
((φ (ψ χ)) → θ)       ((φ ((ψ τ) χ)) → θ)

Theoremadantrrl 455 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
((φ (ψ χ)) → θ)       ((φ (ψ (τ χ))) → θ)

Theoremadantrrr 456 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
((φ (ψ χ)) → θ)       ((φ (ψ (χ τ))) → θ)

Theoremad2antrr 457 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
(φψ)       (((φ χ) θ) → ψ)

Theoremad2antlr 458 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
(φψ)       (((χ φ) θ) → ψ)

Theoremad2antrl 459 Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(φψ)       ((χ (φ θ)) → ψ)

(φψ)       ((χ (θ φ)) → ψ)

Theoremad3antrrr 461 Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
(φψ)       ((((φ χ) θ) τ) → ψ)

Theoremad3antlr 462 Deduction adding three conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φψ)       ((((χ φ) θ) τ) → ψ)

Theoremad4antr 463 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φψ)       (((((φ χ) θ) τ) η) → ψ)

Theoremad4antlr 464 Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φψ)       (((((χ φ) θ) τ) η) → ψ)

Theoremad5antr 465 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φψ)       ((((((φ χ) θ) τ) η) ζ) → ψ)

Theoremad5antlr 466 Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φψ)       ((((((χ φ) θ) τ) η) ζ) → ψ)

Theoremad6antr 467 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φψ)       (((((((φ χ) θ) τ) η) ζ) σ) → ψ)

Theoremad6antlr 468 Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φψ)       (((((((χ φ) θ) τ) η) ζ) σ) → ψ)

Theoremad7antr 469 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φψ)       ((((((((φ χ) θ) τ) η) ζ) σ) ρ) → ψ)

Theoremad7antlr 470 Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φψ)       ((((((((χ φ) θ) τ) η) ζ) σ) ρ) → ψ)

Theoremad8antr 471 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φψ)       (((((((((φ χ) θ) τ) η) ζ) σ) ρ) μ) → ψ)

Theoremad8antlr 472 Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φψ)       (((((((((χ φ) θ) τ) η) ζ) σ) ρ) μ) → ψ)

Theoremad9antr 473 Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φψ)       ((((((((((φ χ) θ) τ) η) ζ) σ) ρ) μ) λ) → ψ)

Theoremad9antlr 474 Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φψ)       ((((((((((χ φ) θ) τ) η) ζ) σ) ρ) μ) λ) → ψ)

Theoremad10antr 475 Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φψ)       (((((((((((φ χ) θ) τ) η) ζ) σ) ρ) μ) λ) κ) → ψ)

Theoremad10antlr 476 Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φψ)       (((((((((((χ φ) θ) τ) η) ζ) σ) ρ) μ) λ) κ) → ψ)

Theoremad2ant2l 477 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ) → χ)       (((θ φ) (τ ψ)) → χ)

Theoremad2ant2r 478 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ) → χ)       (((φ θ) (ψ τ)) → χ)

Theoremad2ant2lr 479 Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
((φ ψ) → χ)       (((θ φ) (ψ τ)) → χ)

Theoremad2ant2rl 480 Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
((φ ψ) → χ)       (((φ θ) (τ ψ)) → χ)

Theoremsimpll 481 Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
(((φ ψ) χ) → φ)

Theoremsimplr 482 Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
(((φ ψ) χ) → ψ)

Theoremsimprl 483 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((φ (ψ χ)) → ψ)

Theoremsimprr 484 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((φ (ψ χ)) → χ)

Theoremsimplll 485 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((φ ψ) χ) θ) → φ)

Theoremsimpllr 486 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((φ ψ) χ) θ) → ψ)

Theoremsimplrl 487 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((φ (ψ χ)) θ) → ψ)

Theoremsimplrr 488 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((φ (ψ χ)) θ) → χ)

Theoremsimprll 489 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((φ ((ψ χ) θ)) → ψ)

Theoremsimprlr 490 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((φ ((ψ χ) θ)) → χ)

Theoremsimprrl 491 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((φ (ψ (χ θ))) → χ)

Theoremsimprrr 492 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((φ (ψ (χ θ))) → θ)

Theorempm4.87 493 Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
(((((φ ψ) → χ) ↔ (φ → (ψχ))) ((φ → (ψχ)) ↔ (ψ → (φχ)))) ((ψ → (φχ)) ↔ ((ψ φ) → χ)))

Theoremabai 494 Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
((φ ψ) ↔ (φ (φψ)))

Theoreman12 495 Swap two conjuncts. Note that the first digit (1) in the label refers to the outer conjunct position, and the next digit (2) to the inner conjunct position. (Contributed by NM, 12-Mar-1995.)
((φ (ψ χ)) ↔ (ψ (φ χ)))

Theoreman32 496 A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)
(((φ ψ) χ) ↔ ((φ χ) ψ))

Theoreman13 497 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
((φ (ψ χ)) ↔ (χ (ψ φ)))

Theoreman31 498 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
(((φ ψ) χ) ↔ ((χ ψ) φ))

Theoreman12s 499 Swap two conjuncts in antecedent. The label suffix "s" means that an12 495 is combined with syl 14 (or a variant). (Contributed by NM, 13-Mar-1996.)
((φ (ψ χ)) → θ)       ((ψ (φ χ)) → θ)

Theoremancom2s 500 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((φ (ψ χ)) → θ)       ((φ (χ ψ)) → θ)

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