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Theorem List for New Foundations Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempm4.25 501 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(φ ↔ (φ φ))

Theoremorim12i 502 Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)
(φψ)    &   (χθ)       ((φ χ) → (ψ θ))

Theoremorim1i 503 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(φψ)       ((φ χ) → (ψ χ))

Theoremorim2i 504 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(φψ)       ((χ φ) → (χ ψ))

Theoremorbi2i 505 Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.)
(φψ)       ((χ φ) ↔ (χ ψ))

Theoremorbi1i 506 Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(φψ)       ((φ χ) ↔ (ψ χ))

Theoremorbi12i 507 Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χθ)       ((φ χ) ↔ (ψ θ))

Theorempm1.5 508 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
((φ (ψ χ)) → (ψ (φ χ)))

Theoremor12 509 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
((φ (ψ χ)) ↔ (ψ (φ χ)))

Theoremorass 510 Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((φ ψ) χ) ↔ (φ (ψ χ)))

Theorempm2.31 511 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((φ (ψ χ)) → ((φ ψ) χ))

Theorempm2.32 512 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
(((φ ψ) χ) → (φ (ψ χ)))

Theoremor32 513 A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((φ ψ) χ) ↔ ((φ χ) ψ))

Theoremor4 514 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
(((φ ψ) (χ θ)) ↔ ((φ χ) (ψ θ)))

Theoremor42 515 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
(((φ ψ) (χ θ)) ↔ ((φ χ) (θ ψ)))

Theoremorordi 516 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
((φ (ψ χ)) ↔ ((φ ψ) (φ χ)))

Theoremorordir 517 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
(((φ ψ) χ) ↔ ((φ χ) (ψ χ)))

Theoremjca 518 Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). Equivalent to the natural deduction rule I ( introduction), see natded in set.mm. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
(φψ)    &   (φχ)       (φ → (ψ χ))

Theoremjcad 519 Deduction conjoining the consequents of two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
(φ → (ψχ))    &   (φ → (ψθ))       (φ → (ψ → (χ θ)))

Theoremjca31 520 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
(φψ)    &   (φχ)    &   (φθ)       (φ → ((ψ χ) θ))

Theoremjca32 521 Join three consequents. (Contributed by FL, 1-Aug-2009.)
(φψ)    &   (φχ)    &   (φθ)       (φ → (ψ (χ θ)))

Theoremjcai 522 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (φ → (ψχ))       (φ → (ψ χ))

Theoremjctil 523 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(φψ)    &   χ       (φ → (χ ψ))

Theoremjctir 524 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(φψ)    &   χ       (φ → (ψ χ))

Theoremjctl 525 Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
ψ       (φ → (ψ φ))

Theoremjctr 526 Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
ψ       (φ → (φ ψ))

Theoremjctild 527 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(φ → (ψχ))    &   (φθ)       (φ → (ψ → (θ χ)))

Theoremjctird 528 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(φ → (ψχ))    &   (φθ)       (φ → (ψ → (χ θ)))

Theoremancl 529 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
((φψ) → (φ → (φ ψ)))

Theoremanclb 530 Conjoin antecedent to left of consequent. Theorem *4.7 of [WhiteheadRussell] p. 120. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((φψ) ↔ (φ → (φ ψ)))

Theorempm5.42 531 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((φ → (ψχ)) ↔ (φ → (ψ → (φ χ))))

Theoremancr 532 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
((φψ) → (φ → (ψ φ)))

Theoremancrb 533 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((φψ) ↔ (φ → (ψ φ)))

Theoremancli 534 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
(φψ)       (φ → (φ ψ))

Theoremancri 535 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
(φψ)       (φ → (ψ φ))

Theoremancld 536 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
(φ → (ψχ))       (φ → (ψ → (ψ χ)))

Theoremancrd 537 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
(φ → (ψχ))       (φ → (ψ → (χ ψ)))

Theoremanc2l 538 Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.)
((φ → (ψχ)) → (φ → (ψ → (φ χ))))

Theoremanc2r 539 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
((φ → (ψχ)) → (φ → (ψ → (χ φ))))

Theoremanc2li 540 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
(φ → (ψχ))       (φ → (ψ → (φ χ)))

Theoremanc2ri 541 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
(φ → (ψχ))       (φ → (ψ → (χ φ)))

Theorempm3.41 542 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((φχ) → ((φ ψ) → χ))

Theorempm3.42 543 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((ψχ) → ((φ ψ) → χ))

Theorempm3.4 544 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
((φ ψ) → (φψ))

Theorempm4.45im 545 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
(φ ↔ (φ (ψφ)))

Theoremanim12d 546 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)
(φ → (ψχ))    &   (φ → (θτ))       (φ → ((ψ θ) → (χ τ)))

Theoremanim1d 547 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)
(φ → (ψχ))       (φ → ((ψ θ) → (χ θ)))

Theoremanim2d 548 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))       (φ → ((θ ψ) → (θ χ)))

Theoremanim12i 549 Conjoin antecedents and consequents of two premises. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.)
(φψ)    &   (χθ)       ((φ χ) → (ψ θ))

Theoremanim12ci 550 Variant of anim12i 549 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φψ)    &   (χθ)       ((φ χ) → (θ ψ))

Theoremanim1i 551 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(φψ)       ((φ χ) → (ψ χ))

Theoremanim2i 552 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(φψ)       ((χ φ) → (χ ψ))

Theoremanim12ii 553 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)
(φ → (ψχ))    &   (θ → (ψτ))       ((φ θ) → (ψ → (χ τ)))

Theoremprth 554 Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 546. Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it praeclarum theorema (splendid theorem). (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
(((φψ) (χθ)) → ((φ χ) → (ψ θ)))

Theorempm2.3 555 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((φ (ψ χ)) → (φ (χ ψ)))

Theorempm2.41 556 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((ψ (φ ψ)) → (φ ψ))

Theorempm2.42 557 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((¬ φ (φψ)) → (φψ))

Theorempm2.4 558 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((φ (φ ψ)) → (φ ψ))

Theorempm2.65da 559 Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
((φ ψ) → χ)    &   ((φ ψ) → ¬ χ)       (φ → ¬ ψ)

Theorempm4.44 560 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(φ ↔ (φ (φ ψ)))

Theorempm4.14 561 Theorem *4.14 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
(((φ ψ) → χ) ↔ ((φ ¬ χ) → ¬ ψ))

Theorempm3.37 562 Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
(((φ ψ) → χ) → ((φ ¬ χ) → ¬ ψ))

Theoremnan 563 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
((φ → ¬ (ψ χ)) ↔ ((φ ψ) → ¬ χ))

Theorempm4.15 564 Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
(((φ ψ) → ¬ χ) ↔ ((ψ χ) → ¬ φ))

Theorempm4.78 565 Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
(((φψ) (φχ)) ↔ (φ → (ψ χ)))

Theorempm4.79 566 Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)
(((ψφ) (χφ)) ↔ ((ψ χ) → φ))

Theorempm4.87 567 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)))

Theorempm3.33 568 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((φψ) (ψχ)) → (φχ))

Theorempm3.34 569 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((ψχ) (φψ)) → (φχ))

Theorempm3.35 570 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)
((φ (φψ)) → ψ)

Theorempm5.31 571 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((χ (φψ)) → (φ → (ψ χ)))

Theoremimp4a 572 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       (φ → (ψ → ((χ θ) → τ)))

Theoremimp4b 573 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       ((φ ψ) → ((χ θ) → τ))

Theoremimp4c 574 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       (φ → (((ψ χ) θ) → τ))

Theoremimp4d 575 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       (φ → ((ψ (χ θ)) → τ))

Theoremimp41 576 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       ((((φ ψ) χ) θ) → τ)

Theoremimp42 577 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       (((φ (ψ χ)) θ) → τ)

Theoremimp43 578 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       (((φ ψ) (χ θ)) → τ)

Theoremimp44 579 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       ((φ ((ψ χ) θ)) → τ)

Theoremimp45 580 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       ((φ (ψ (χ θ))) → τ)

Theoremimp5a 581 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → (ψ → (χ → (θ → (τη)))))       (φ → (ψ → (χ → ((θ τ) → η))))

Theoremimp5d 582 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → (ψ → (χ → (θ → (τη)))))       (((φ ψ) χ) → ((θ τ) → η))

Theoremimp5g 583 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → (ψ → (χ → (θ → (τη)))))       ((φ ψ) → (((χ θ) τ) → η))

Theoremimp55 584 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → (ψ → (χ → (θ → (τη)))))       (((φ (ψ (χ θ))) τ) → η)

Theoremimp511 585 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → (ψ → (χ → (θ → (τη)))))       ((φ ((ψ (χ θ)) τ)) → η)

Theoremexpimpd 586 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
((φ ψ) → (χθ))       (φ → ((ψ χ) → θ))

Theoremexp31 587 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((φ ψ) χ) → θ)       (φ → (ψ → (χθ)))

Theoremexp32 588 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((φ (ψ χ)) → θ)       (φ → (ψ → (χθ)))

Theoremexp4a 589 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → ((χ θ) → τ)))       (φ → (ψ → (χ → (θτ))))

Theoremexp4b 590 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
((φ ψ) → ((χ θ) → τ))       (φ → (ψ → (χ → (θτ))))

Theoremexp4c 591 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (((ψ χ) θ) → τ))       (φ → (ψ → (χ → (θτ))))

Theoremexp4d 592 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(φ → ((ψ (χ θ)) → τ))       (φ → (ψ → (χ → (θτ))))

Theoremexp41 593 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((((φ ψ) χ) θ) → τ)       (φ → (ψ → (χ → (θτ))))

Theoremexp42 594 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((φ (ψ χ)) θ) → τ)       (φ → (ψ → (χ → (θτ))))

Theoremexp43 595 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((φ ψ) (χ θ)) → τ)       (φ → (ψ → (χ → (θτ))))

Theoremexp44 596 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((φ ((ψ χ) θ)) → τ)       (φ → (ψ → (χ → (θτ))))

Theoremexp45 597 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((φ (ψ (χ θ))) → τ)       (φ → (ψ → (χ → (θτ))))

Theoremexpr 598 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((φ (ψ χ)) → θ)       ((φ ψ) → (χθ))

Theoremexp5c 599 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → ((ψ χ) → ((θ τ) → η)))       (φ → (ψ → (χ → (θ → (τη)))))

Theoremexp53 600 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
((((φ ψ) (χ θ)) τ) → η)       (φ → (ψ → (χ → (θ → (τη)))))

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