HomeHome Intuitionistic Logic Explorer
Theorem List (p. 7 of 114)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 601-700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm2.65i 601 Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)        ¬ 𝜑
 
Theoremmt2 602 A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)
𝜓    &   (𝜑 → ¬ 𝜓)        ¬ 𝜑
 
Theorembiijust 603 Theorem used to justify definition of intuitionistic biconditional df-bi 115. (Contributed by NM, 24-Nov-2017.)
((((𝜑𝜓) ∧ (𝜓𝜑)) → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ (((𝜑𝜓) ∧ (𝜓𝜑)) → ((𝜑𝜓) ∧ (𝜓𝜑))))
 
Theoremcon3 604 Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 
Theoremcon2 605 Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑))
 
Theoremmt2i 606 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
𝜒    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremnotnoti 607 Infer double negation. (Contributed by NM, 27-Feb-2008.)
𝜑        ¬ ¬ 𝜑
 
Theorempm2.21i 608 A contradiction implies anything. Inference from pm2.21 580. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ 𝜑       (𝜑𝜓)
 
Theorempm2.24ii 609 A contradiction implies anything. Inference from pm2.24 584. (Contributed by NM, 27-Feb-2008.)
𝜑    &    ¬ 𝜑       𝜓
 
Theoremnsyld 610 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
(𝜑 → (𝜓 → ¬ 𝜒))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜓 → ¬ 𝜏))
 
Theoremnsyli 611 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → ¬ 𝜒)       (𝜑 → (𝜃 → ¬ 𝜓))
 
Theoremmth8 612 Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))
 
Theoremjc 613 Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → ¬ (𝜓 → ¬ 𝜒))
 
Theorempm2.51 614 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 → ¬ 𝜓))
 
Theorempm2.52 615 Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
(¬ (𝜑𝜓) → (¬ 𝜑 → ¬ 𝜓))
 
Theoremexpt 616 Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓𝜒)))
 
Theoremjarl 617 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
(((𝜑𝜓) → 𝜒) → (¬ 𝜑𝜒))
 
Theorempm2.65 618 Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here 𝜑, derive a contradiction, and therefore conclude ¬ 𝜑, are valid intuitionistically (and can be called "proof of negation", for example by Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume ¬ 𝜑, derive a contradiction, and conclude 𝜑, such as condandc 811, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
((𝜑𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))
 
Theorempm2.65d 619 Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)
 
Theorempm2.65da 620 Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓) → ¬ 𝜒)       (𝜑 → ¬ 𝜓)
 
Theoremmto 621 The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
¬ 𝜓    &   (𝜑𝜓)        ¬ 𝜑
 
Theoremmtod 622 Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremmtoi 623 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremmtand 624 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑 → ¬ 𝜒)    &   ((𝜑𝜓) → 𝜒)       (𝜑 → ¬ 𝜓)
 
Theoremnotbid 625 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
 
Theoremcon2b 626 Contraposition. Bidirectional version of con2 605. (Contributed by NM, 5-Aug-1993.)
((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
 
Theoremnotbii 627 Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑𝜓)       𝜑 ↔ ¬ 𝜓)
 
Theoremmtbi 628 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
¬ 𝜑    &   (𝜑𝜓)        ¬ 𝜓
 
Theoremmtbir 629 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)
¬ 𝜓    &   (𝜑𝜓)        ¬ 𝜑
 
Theoremmtbid 630 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)
 
Theoremmtbird 631 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremmtbii 632 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
¬ 𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)
 
Theoremmtbiri 633 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremsylnib 634 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 → ¬ 𝜒)
 
Theoremsylnibr 635 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 636 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 637 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜓𝜑)    &   𝜓𝜒)       𝜑𝜒)
 
Theoremxchnxbi 638 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜑𝜒)       𝜒𝜓)
 
Theoremxchnxbir 639 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜒𝜑)       𝜒𝜓)
 
Theoremxchbinx 640 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 ↔ ¬ 𝜒)
 
Theoremxchbinxr 641 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 ↔ ¬ 𝜒)
 
Theoremmt2bi 642 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 643 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))
 
Theorempm5.21 644 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 645 Two propositions are equivalent if they are both false. Closed form of 2false 650. Equivalent to a bi2 128-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑 → (¬ 𝜓 → (𝜑𝜓)))
 
Theoremnbn2 646 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 647 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))
 
Theoremnbn 648 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 649 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
𝜑       𝜓 ↔ (𝜓 ↔ ¬ 𝜑))
 
Theorem2false 650 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ 𝜑    &    ¬ 𝜓       (𝜑𝜓)
 
Theorem2falsed 651 Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓𝜒))
 
Theorempm5.21ni 652 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 653 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 654 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))
 
Theorempm5.19 655 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ (𝜑 ↔ ¬ 𝜑)
 
Theorempm4.8 656 Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 850 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
 
Theoremimnan 657 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremimnani 658 Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
¬ (𝜑𝜓)       (𝜑 → ¬ 𝜓)
 
Theoremnan 659 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))
 
Theorempm3.24 660 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 661 Triple negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
(¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)
 
1.2.6  Logical disjunction
 
Syntaxwo 662 Extend wff definition to include disjunction ('or').
wff (𝜑𝜓)
 
Axiomax-io 663 Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 664 instead. (New usage is discouraged.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
 
Theoremjaob 664 Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 663. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
 
Theoremolc 665 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 666 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))
 
Theorempm2.67-2 667 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))
 
Theorempm3.44 668 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 669 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
(𝜑𝜓)    &   (𝜒𝜓)       ((𝜑𝜒) → 𝜓)
 
Theoremjaod 670 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 671 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)
 
Theoremjaao 672 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))
 
Theoremjaoa 673 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))
 
Theorempm2.53 674 Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 826). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
((𝜑𝜓) → (¬ 𝜑𝜓))
 
Theoremori 675 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑𝜓)       𝜑𝜓)
 
Theoremord 676 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓𝜒))
 
Theoremorel1 677 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 678 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))
 
Theorempm1.4 679 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremorcom 680 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 681 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))
 
Theoremorcoms 682 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremorci 683 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑       (𝜑𝜓)
 
Theoremolci 684 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑       (𝜓𝜑)
 
Theoremorcd 685 Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒))
 
Theoremolcd 686 Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))
 
Theoremorcs 687 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 688 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)
 
Theorempm2.07 689 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (𝜑𝜑))
 
Theorempm2.45 690 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜑)
 
Theorempm2.46 691 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜓)
 
Theorempm2.47 692 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑𝜓))
 
Theorempm2.48 693 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 ∨ ¬ 𝜓))
 
Theorempm2.49 694 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
 
Theorempm2.67 695 Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
(((𝜑𝜓) → 𝜓) → (𝜑𝜓))
 
Theorembiorf 696 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 697 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
(𝜑 → (𝜓 ↔ (¬ 𝜑𝜓)))
 
Theorembiorfi 698 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
¬ 𝜑       (𝜓 ↔ (𝜓𝜑))
 
Theorempm2.621 699 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))
 
Theorempm2.62 700 Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11394
  Copyright terms: Public domain < Previous  Next >