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Theorem List for Intuitionistic Logic Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpancom 401 An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
(ψφ)    &   ((φ ψ) → χ)       (ψχ)
 
Theoremmpan 402 An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
φ    &   ((φ ψ) → χ)       (ψχ)
 
Theoremmpan2 403 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 404 An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
φ    &   ψ    &   ((φ ψ) → χ)       χ
 
Theoremmp4an 405 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2011.)
φ    &   ψ    &   χ    &   θ    &   (((φ ψ) (χ θ)) → τ)       τ
 
Theoremmpan2d 406 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(φχ)    &   (φ → ((ψ χ) → θ))       (φ → (ψθ))
 
Theoremmpand 407 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 408 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 409 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 410 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
ψ    &   χ    &   (φ → ((ψ χ) → θ))       (φθ)
 
Theoremmp2and 411 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
(φψ)    &   (φχ)    &   (φ → ((ψ χ) → θ))       (φθ)
 
Theoremmpanl1 412 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 413 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 414 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
φ    &   ψ    &   (((φ ψ) χ) → θ)       (χθ)
 
Theoremmpanr1 415 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 416 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)
 
Theoremmpanr2OLD 417 Obsolete proof of mpanr2 416 as of 7-Apr-2013. (Contributed by NM, 3-May-1994.) (Revised by NM, 12-May-2011.)
χ    &   ((φ (ψ χ)) → θ)       ((φ ψ) → θ)
 
Theoremmpanr12 418 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
ψ    &   χ    &   ((φ (ψ χ)) → θ)       (φθ)
 
Theoremmpanlr1 419 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 420 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)
((φ ψ) → (χθ))       (φ → ((ψχ) ↔ (ψθ)))
 
Theoremimdistan 421 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 422 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
(φ → (ψχ))       ((φ ψ) → (φ χ))
 
Theoremimdistanri 423 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
(φ → (ψχ))       ((ψ φ) → (χ φ))
 
Theoremimdistand 424 Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.)
(φ → (ψ → (χθ)))       (φ → ((ψ χ) → (ψ θ)))
 
Theoremimdistanda 425 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 426 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 427 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.)
(φ → (ψ → (χθ)))       (φ → ((χ ψ) ↔ (θ ψ)))
 
Theorempm5.32da 428 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.)
((φ ψ) → (χθ))       (φ → ((ψ χ) ↔ (ψ θ)))
 
Theorempm5.32 429 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 430 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
(φ → (ψχ))       ((φ ψ) ↔ (φ χ))
 
Theorempm5.32ri 431 Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)
(φ → (ψχ))       ((ψ φ) ↔ (χ φ))
 
Theorembiadan2 432 Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
(φψ)    &   (ψ → (φχ))       (φ ↔ (ψ χ))
 
Theoremanbi2i 433 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 434 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 435 Variant of anbi2i 433 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(φψ)       ((φ χ) ↔ (χ ψ))
 
Theoremanbi12i 436 Conjoin both sides of two equivalences. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χθ)       ((φ χ) ↔ (ψ θ))
 
Theoremanbi12ci 437 Variant of anbi12i 436 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φψ)    &   (χθ)       ((φ χ) ↔ (θ ψ))
 
Theoremsylan9bb 438 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(φ → (ψχ))    &   (θ → (χτ))       ((φ θ) → (ψτ))
 
Theoremsylan9bbr 439 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(φ → (ψχ))    &   (θ → (χτ))       ((θ φ) → (ψτ))
 
Theoremanbi2d 440 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 441 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 442 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 443 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
((φψ) → ((χ φ) ↔ (χ ψ)))
 
Theorembitr 444 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(((φψ) (ψχ)) → (φχ))
 
Theoremanbi12d 445 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 446 Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)
((((φψ) χ) φ) → (ψ χ))
 
Theorempm5.3 447 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 448 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 449 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 450 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 451 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 452 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 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)
 
Theoremadantlrl 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 (τ ψ)) χ) → θ)
 
Theoremadantlrr 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)
 
Theoremadantrll 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 ((τ ψ) χ)) → θ)
 
Theoremadantrlr 457 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 458 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 459 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 460 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
(φψ)       (((φ χ) θ) → ψ)
 
Theoremad3antrrr 461 Deduction adding 3 conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
(φψ)       ((((φ χ) θ) τ) → ψ)
 
Theoremad2antlr 462 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
(φψ)       (((χ φ) θ) → ψ)
 
Theoremad2antrl 463 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(φψ)       ((χ (φ θ)) → ψ)
 
Theoremad2antll 464 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
(φψ)       ((χ (θ φ)) → ψ)
 
Theoremad2ant2l 465 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ) → χ)       (((θ φ) (τ ψ)) → χ)
 
Theoremad2ant2r 466 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ) → χ)       (((φ θ) (ψ τ)) → χ)
 
Theoremad2ant2lr 467 Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
((φ ψ) → χ)       (((θ φ) (ψ τ)) → χ)
 
Theoremad2ant2rl 468 Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
((φ ψ) → χ)       (((φ θ) (τ ψ)) → χ)
 
Theoremsimpll 469 Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
(((φ ψ) χ) → φ)
 
Theoremsimplr 470 Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
(((φ ψ) χ) → ψ)
 
Theoremsimprl 471 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((φ (ψ χ)) → ψ)
 
Theoremsimprr 472 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((φ (ψ χ)) → χ)
 
Theoremsimplll 473 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((φ ψ) χ) θ) → φ)
 
Theoremsimpllr 474 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((φ ψ) χ) θ) → ψ)
 
Theoremsimplrl 475 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((φ (ψ χ)) θ) → ψ)
 
Theoremsimplrr 476 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((φ (ψ χ)) θ) → χ)
 
Theoremsimprll 477 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((φ ((ψ χ) θ)) → ψ)
 
Theoremsimprlr 478 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((φ ((ψ χ) θ)) → χ)
 
Theoremsimprrl 479 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((φ (ψ (χ θ))) → χ)
 
Theoremsimprrr 480 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((φ (ψ (χ θ))) → θ)
 
Theorempm4.87 481 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 482 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 483 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 484 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 485 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 486 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 487 Swap two conjuncts in antecedent. The label suffix "s" means that an12 483 is combined with syl 14 (or a variant). (Contributed by NM, 13-Mar-1996.)
((φ (ψ χ)) → θ)       ((ψ (φ χ)) → θ)
 
Theoremancom2s 488 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)
 
Theoreman13s 489 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
((φ (ψ χ)) → θ)       ((χ (ψ φ)) → θ)
 
Theoreman32s 490 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
(((φ ψ) χ) → θ)       (((φ χ) ψ) → θ)
 
Theoremancom1s 491 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)       (((ψ φ) χ) → θ)
 
Theoreman31s 492 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
(((φ ψ) χ) → θ)       (((χ ψ) φ) → θ)
 
Theoremanabs1 493 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(((φ ψ) φ) ↔ (φ ψ))
 
Theoremanabs5 494 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
((φ (φ ψ)) ↔ (φ ψ))
 
Theoremanabs7 495 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
((ψ (φ ψ)) ↔ (φ ψ))
 
Theoremanabsan 496 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
(((φ φ) ψ) → χ)       ((φ ψ) → χ)
 
Theoremanabss1 497 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
(((φ ψ) φ) → χ)       ((φ ψ) → χ)
 
Theoremanabss4 498 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
(((ψ φ) ψ) → χ)       ((φ ψ) → χ)
 
Theoremanabss5 499 Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
((φ (φ ψ)) → χ)       ((φ ψ) → χ)
 
Theoremanabsi5 500 Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
(φ → ((φ ψ) → χ))       ((φ ψ) → χ)
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