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Theorem List for Intuitionistic Logic Explorer - 601-700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembibif 601 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
ψ → ((φψ) ↔ ¬ φ))

Theoremnbn 602 The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
¬ φ       ψ ↔ (ψφ))

Theoremnbn3 603 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
φ       ψ ↔ (ψ ↔ ¬ φ))

Theorem2false 604 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ φ    &    ¬ ψ       (φψ)

Theorem2falsed 605 Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
(φ → ¬ ψ)    &   (φ → ¬ χ)       (φ → (ψχ))

Theorempm5.21ni 606 Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)
(φψ)    &   (χψ)       ψ → (φχ))

Theorempm5.21nii 607 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φψ)    &   (χψ)    &   (ψ → (φχ))       (φχ)

Theorempm5.21ndd 608 Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φ → (χψ))    &   (φ → (θψ))    &   (φ → (ψ → (χθ)))       (φ → (χθ))

Theorempm5.19 609 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ (φ ↔ ¬ φ)

Theorempm4.8 610 Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 802 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
((φ → ¬ φ) ↔ ¬ φ)

Theoremimnan 611 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
((φ → ¬ ψ) ↔ ¬ (φ ψ))

Theoremimnani 612 Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
¬ (φ ψ)       (φ → ¬ ψ)

Theoremnan 613 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
((φ → ¬ (ψ χ)) ↔ ((φ ψ) → ¬ χ))

Theorempm3.24 614 Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 2-Feb-2015.)
¬ (φ ¬ φ)

Theoremnotnotnot 615 Triple negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
(¬ ¬ ¬ φ ↔ ¬ φ)

1.2.6  Logical disjunction

Syntaxwo 616 Extend wff definition to include disjunction ('or').
wff (φ ψ)

Axiomax-io 617 Definition of 'or'. Axiom 6 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
(((φ χ) → ψ) ↔ ((φψ) (χψ)))

Theoremjaob 618 Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(((φ χ) → ψ) ↔ ((φψ) (χψ)))

Theoremolc 619 Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
(φ → (ψ φ))

Theoremorc 620 Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
(φ → (φ ψ))

Theorempm2.67-2 621 Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
(((φ χ) → ψ) → (φψ))

Theorempm3.44 622 Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(((ψφ) (χφ)) → ((ψ χ) → φ))

Theoremjaoi 623 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
(φψ)    &   (χψ)       ((φ χ) → ψ)

Theoremjaod 624 Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) (Revised by NM, 4-Apr-2013.)
(φ → (ψχ))    &   (φ → (θχ))       (φ → ((ψ θ) → χ))

Theoremmpjaod 625 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
(φ → (ψχ))    &   (φ → (θχ))    &   (φ → (ψ θ))       (φχ)

Theoremjaao 626 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
(φ → (ψχ))    &   (θ → (τχ))       ((φ θ) → ((ψ τ) → χ))

Theoremjaoa 627 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
(φ → (ψχ))    &   (θ → (τχ))       ((φ θ) → ((ψ τ) → χ))

Theorempm2.53 628 Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 778). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
((φ ψ) → (¬ φψ))

Theoremori 629 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φ ψ)       φψ)

Theoremord 630 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φ → (ψ χ))       (φ → (¬ ψχ))

Theoremorel1 631 Elimination of disjunction by denial of a disjunct. Theorem *2.55 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 21-Jul-2012.)
φ → ((φ ψ) → ψ))

Theoremorel2 632 Elimination of disjunction by denial of a disjunct. Theorem *2.56 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Apr-2013.)
φ → ((ψ φ) → ψ))

Theorempm1.4 633 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
((φ ψ) → (ψ φ))

Theoremorcom 634 Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.)
((φ ψ) ↔ (ψ φ))

Theoremorcomd 635 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
(φ → (ψ χ))       (φ → (χ ψ))

Theoremorcoms 636 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
((φ ψ) → χ)       ((ψ φ) → χ)

Theoremorci 637 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
φ       (φ ψ)

Theoremolci 638 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
φ       (ψ φ)

Theoremorcd 639 Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
(φψ)       (φ → (ψ χ))

Theoremolcd 640 Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(φψ)       (φ → (χ ψ))

Theoremorcs 641 Deduction eliminating disjunct. 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 14) -type inference in a proof. (Contributed by NM, 21-Jun-1994.)
((φ ψ) → χ)       (φχ)

Theoremolcs 642 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
((φ ψ) → χ)       (ψχ)

Theorempm2.07 643 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(φ → (φ φ))

Theorempm2.45 644 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (φ ψ) → ¬ φ)

Theorempm2.46 645 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (φ ψ) → ¬ ψ)

Theorempm2.47 646 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (φ ψ) → (¬ φ ψ))

Theorempm2.48 647 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (φ ψ) → (φ ¬ ψ))

Theorempm2.49 648 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (φ ψ) → (¬ φ ¬ ψ))

Theorempm2.67 649 Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
(((φ ψ) → ψ) → (φψ))

Theorembiorf 650 A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
φ → (ψ ↔ (φ ψ)))

Theorembiortn 651 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
(φ → (ψ ↔ (¬ φ ψ)))

Theorembiorfi 652 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
¬ φ       (ψ ↔ (ψ φ))

Theorempm2.621 653 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
((φψ) → ((φ ψ) → ψ))

Theorempm2.62 654 Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
((φ ψ) → ((φψ) → ψ))

Theoremimorri 655 Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.)
φ ψ)       (φψ)

Theoremioran 656 Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 795, anordc 849, or ianordc 787. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(¬ (φ ψ) ↔ (¬ φ ¬ ψ))

Theorempm3.14 657 Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 787. The converse holds for decidable propositions, as seen at pm3.13dc 852. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
((¬ φ ¬ ψ) → ¬ (φ ψ))

Theorempm3.1 658 Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 849. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
((φ ψ) → ¬ (¬ φ ¬ ψ))

Theoremjao 659 Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 4-Apr-2013.)
((φψ) → ((χψ) → ((φ χ) → ψ)))

Theorempm1.2 660 Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.)
((φ φ) → φ)

Theoremoridm 661 Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.)
((φ φ) ↔ φ)

Theorempm4.25 662 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(φ ↔ (φ φ))

Theoremorim12i 663 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 664 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(φψ)       ((φ χ) → (ψ χ))

Theoremorim2i 665 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(φψ)       ((χ φ) → (χ ψ))

Theoremorbi2i 666 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 667 Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(φψ)       ((φ χ) ↔ (ψ χ))

Theoremorbi12i 668 Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χθ)       ((φ χ) ↔ (ψ θ))

Theorempm1.5 669 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
((φ (ψ χ)) → (ψ (φ χ)))

Theoremor12 670 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
((φ (ψ χ)) ↔ (ψ (φ χ)))

Theoremorass 671 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)))

Theorempm2.31 672 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((φ (ψ χ)) → ((φ ψ) χ))

Theorempm2.32 673 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
(((φ ψ) χ) → (φ (ψ χ)))

Theoremor32 674 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 675 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
(((φ ψ) (χ θ)) ↔ ((φ χ) (ψ θ)))

Theoremor42 676 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
(((φ ψ) (χ θ)) ↔ ((φ χ) (θ ψ)))

Theoremorordi 677 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
((φ (ψ χ)) ↔ ((φ ψ) (φ χ)))

Theoremorordir 678 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
(((φ ψ) χ) ↔ ((φ χ) (ψ χ)))

Theorempm2.3 679 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((φ (ψ χ)) → (φ (χ ψ)))

Theorempm2.41 680 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((ψ (φ ψ)) → (φ ψ))

Theorempm2.42 681 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((¬ φ (φψ)) → (φψ))

Theorempm2.4 682 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((φ (φ ψ)) → (φ ψ))

Theorempm4.44 683 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(φ ↔ (φ (φ ψ)))

Theoremmtord 684 A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φ → ¬ χ)    &   (φ → ¬ θ)    &   (φ → (ψ → (χ θ)))       (φ → ¬ ψ)

Theorempm4.45 685 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(φ ↔ (φ (φ ψ)))

Theorempm3.48 686 Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) (Revised by NM, 1-Dec-2012.)
(((φψ) (χθ)) → ((φ χ) → (ψ θ)))

Theoremorim12d 687 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.)
(φ → (ψχ))    &   (φ → (θτ))       (φ → ((ψ θ) → (χ τ)))

Theoremorim1d 688 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
(φ → (ψχ))       (φ → ((ψ θ) → (χ θ)))

Theoremorim2d 689 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
(φ → (ψχ))       (φ → ((θ ψ) → (θ χ)))

Theoremorim2 690 Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
((ψχ) → ((φ ψ) → (φ χ)))

Theoremorbi2d 691 Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φ → (ψχ))       (φ → ((θ ψ) ↔ (θ χ)))

Theoremorbi1d 692 Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))       (φ → ((ψ θ) ↔ (χ θ)))

Theoremorbi1 693 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
((φψ) → ((φ χ) ↔ (ψ χ)))

Theoremorbi12d 694 Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (φ → (θτ))       (φ → ((ψ θ) ↔ (χ τ)))

Theorempm5.61 695 Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.)
(((φ ψ) ¬ ψ) ↔ (φ ¬ ψ))

Theoremjaoian 696 Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
((φ ψ) → χ)    &   ((θ ψ) → χ)       (((φ θ) ψ) → χ)

Theoremjaodan 697 Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
((φ ψ) → χ)    &   ((φ θ) → χ)       ((φ (ψ θ)) → χ)

Theoremmpjaodan 698 Eliminate a disjunction in a deduction. A translation of natural deduction rule E ( elimination). (Contributed by Mario Carneiro, 29-May-2016.)
((φ ψ) → χ)    &   ((φ θ) → χ)    &   (φ → (ψ θ))       (φχ)

Theorempm4.77 699 Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((ψφ) (χφ)) ↔ ((ψ χ) → φ))

Theorempm2.63 700 Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((φ ψ) → ((¬ φ ψ) → ψ))

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