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

Theoremsylnib 601 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 → ¬ 𝜒)

Theoremsylnibr 602 A mixed syllogism inference from an implication and a biconditional. Useful for substituting an consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 → ¬ 𝜒)

Theoremsylnbi 603 A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑𝜓)    &   𝜓𝜒)       𝜑𝜒)

Theoremsylnbir 604 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜓𝜑)    &   𝜓𝜒)       𝜑𝜒)

Theoremxchnxbi 605 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜑𝜒)       𝜒𝜓)

Theoremxchnxbir 606 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜒𝜑)       𝜒𝜓)

Theoremxchbinx 607 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 ↔ ¬ 𝜒)

Theoremxchbinxr 608 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 ↔ ¬ 𝜒)

Theoremmt2bi 609 A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
𝜑       𝜓 ↔ (𝜓 → ¬ 𝜑))

Theoremmtt 610 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))

Theorempm5.21 611 Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑𝜓))

Theorempm5.21im 612 Two propositions are equivalent if they are both false. Closed form of 2false 617. Equivalent to a bi2 121-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑 → (¬ 𝜓 → (𝜑𝜓)))

Theoremnbn2 613 The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑 → (¬ 𝜓 ↔ (𝜑𝜓)))

Theorembibif 614 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))

Theoremnbn 615 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 616 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
𝜑       𝜓 ↔ (𝜓 ↔ ¬ 𝜑))

Theorem2false 617 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ 𝜑    &    ¬ 𝜓       (𝜑𝜓)

Theorem2falsed 618 Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓𝜒))

Theorempm5.21ni 619 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 620 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 621 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))

Theorempm5.19 622 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ (𝜑 ↔ ¬ 𝜑)

Theorempm4.8 623 Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 814 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)

Theoremimnan 624 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))

Theoremimnani 625 Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
¬ (𝜑𝜓)       (𝜑 → ¬ 𝜓)

Theoremnan 626 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))

Theorempm3.24 627 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 628 Triple negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
(¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)

1.2.6  Logical disjunction

Syntaxwo 629 Extend wff definition to include disjunction ('or').
wff (𝜑𝜓)

Axiomax-io 630 Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

Theoremjaob 631 Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

Theoremolc 632 Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
(𝜑 → (𝜓𝜑))

Theoremorc 633 Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
(𝜑 → (𝜑𝜓))

Theorempm2.67-2 634 Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
(((𝜑𝜒) → 𝜓) → (𝜑𝜓))

Theorempm3.44 635 Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(((𝜓𝜑) ∧ (𝜒𝜑)) → ((𝜓𝜒) → 𝜑))

Theoremjaoi 636 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
(𝜑𝜓)    &   (𝜒𝜓)       ((𝜑𝜒) → 𝜓)

Theoremjaod 637 Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) (Revised by NM, 4-Apr-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → ((𝜓𝜃) → 𝜒))

Theoremmpjaod 638 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)

Theoremjaao 639 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))

Theoremjaoa 640 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))

Theorempm2.53 641 Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 790). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
((𝜑𝜓) → (¬ 𝜑𝜓))

Theoremori 642 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑𝜓)       𝜑𝜓)

Theoremord 643 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓𝜒))

Theoremorel1 644 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 → ((𝜑𝜓) → 𝜓))

Theoremorel2 645 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 → ((𝜓𝜑) → 𝜓))

Theorempm1.4 646 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))

Theoremorcom 647 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.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theoremorcomd 648 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))

Theoremorcoms 649 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theoremorci 650 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑       (𝜑𝜓)

Theoremolci 651 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑       (𝜓𝜑)

Theoremorcd 652 Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒))

Theoremolcd 653 Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))

Theoremorcs 654 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.)
((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theoremolcs 655 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theorempm2.07 656 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (𝜑𝜑))

Theorempm2.45 657 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜑)

Theorempm2.46 658 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜓)

Theorempm2.47 659 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑𝜓))

Theorempm2.48 660 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 ∨ ¬ 𝜓))

Theorempm2.49 661 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))

Theorempm2.67 662 Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
(((𝜑𝜓) → 𝜓) → (𝜑𝜓))

Theorembiorf 663 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 ↔ (𝜑𝜓)))

Theorembiortn 664 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
(𝜑 → (𝜓 ↔ (¬ 𝜑𝜓)))

Theorembiorfi 665 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
¬ 𝜑       (𝜓 ↔ (𝜓𝜑))

Theorempm2.621 666 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))

Theorempm2.62 667 Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))

Theoremimorri 668 Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑𝜓)       (𝜑𝜓)

Theoremioran 669 Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 807, anordc 863, or ianordc 799. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))

Theorempm3.14 670 Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 799. The converse holds for decidable propositions, as seen at pm3.13dc 866. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑𝜓))

Theorempm3.1 671 Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 863. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
((𝜑𝜓) → ¬ (¬ 𝜑 ∨ ¬ 𝜓))

Theoremjao 672 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) → ((𝜑𝜒) → 𝜓)))

Theorempm1.2 673 Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.)
((𝜑𝜑) → 𝜑)

Theoremoridm 674 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.)
((𝜑𝜑) ↔ 𝜑)

Theorempm4.25 675 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑𝜑))

Theoremorim12i 676 Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) → (𝜓𝜃))

Theoremorim1i 677 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(𝜑𝜓)       ((𝜑𝜒) → (𝜓𝜒))

Theoremorim2i 678 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))

Theoremorbi2i 679 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)       ((𝜒𝜑) ↔ (𝜒𝜓))

Theoremorbi1i 680 Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))

Theoremorbi12i 681 Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theorempm1.5 682 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → (𝜓 ∨ (𝜑𝜒)))

Theoremor12 683 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
((𝜑 ∨ (𝜓𝜒)) ↔ (𝜓 ∨ (𝜑𝜒)))

Theoremorass 684 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.)
(((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))

Theorempm2.31 685 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) ∨ 𝜒))

Theorempm2.32 686 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∨ 𝜒) → (𝜑 ∨ (𝜓𝜒)))

Theoremor32 687 A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∨ 𝜓))

Theoremor4 688 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜓𝜃)))

Theoremor42 689 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜃𝜓)))

Theoremorordi 690 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))

Theoremorordir 691 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))

Theorempm2.3 692 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → (𝜑 ∨ (𝜒𝜓)))

Theorempm2.41 693 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜓 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm2.42 694 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm2.4 695 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm4.44 696 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∨ (𝜑𝜓)))

Theoremmtord 697 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 → (𝜒𝜃)))       (𝜑 → ¬ 𝜓)

Theorempm4.45 698 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∧ (𝜑𝜓)))

Theorempm3.48 699 Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) (Revised by NM, 1-Dec-2012.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))

Theoremorim12d 700 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))

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