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Theorem List for Metamath Proof Explorer - 801-900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembitr 801 Theorem *4.22 of [WhiteheadRussell] p. 117. bitri 276 in closed form. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
 
Theorembiantr 802 A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
(((𝜑𝜓) ∧ (𝜒𝜓)) → (𝜑𝜒))
 
Theorempm4.14 803 Theorem *4.14 of [WhiteheadRussell] p. 117. Related to con34b 317. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))
 
Theorempm3.37 804 Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
(((𝜑𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))
 
Theoremanim12 805 Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 608. 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)) → ((𝜑𝜒) → (𝜓𝜃)))
 
Theorempm3.4 806 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremexbiri 807 Inference form of exbir 40692. This proof is exbiriVD 41068 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜃𝜒)))
 
Theorempm2.61ian 808 Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)
((𝜑𝜓) → 𝜒)    &   ((¬ 𝜑𝜓) → 𝜒)       (𝜓𝜒)
 
Theorempm2.61dan 809 Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜑 ∧ ¬ 𝜓) → 𝜒)       (𝜑𝜒)
 
Theorempm2.61ddan 810 Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.)
((𝜑𝜓) → 𝜃)    &   ((𝜑𝜒) → 𝜃)    &   ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃)       (𝜑𝜃)
 
Theorempm2.61dda 811 Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.)
((𝜑 ∧ ¬ 𝜓) → 𝜃)    &   ((𝜑 ∧ ¬ 𝜒) → 𝜃)    &   ((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑𝜃)
 
Theoremmtand 812 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑 → ¬ 𝜒)    &   ((𝜑𝜓) → 𝜒)       (𝜑 → ¬ 𝜓)
 
Theorempm2.65da 813 Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓) → ¬ 𝜒)       (𝜑 → ¬ 𝜓)
 
Theoremcondan 814 Proof by contradiction. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Wolf Lammen, 19-Jun-2014.)
((𝜑 ∧ ¬ 𝜓) → 𝜒)    &   ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒)       (𝜑𝜓)
 
Theorembiadan 815 An implication is equivalent to the equivalence of some implied equivalence and some other equivalence involving a conjunction. A utility lemma as illustrated in biadanii 818 and elelb 34111. (Contributed by BJ, 4-Mar-2023.) (Proof shortened by Wolf Lammen, 8-Mar-2023.)
((𝜑𝜓) ↔ ((𝜓 → (𝜑𝜒)) ↔ (𝜑 ↔ (𝜓𝜒))))
 
Theorembiadani 816 Inference associated with biadan 815. (Contributed by BJ, 4-Mar-2023.)
(𝜑𝜓)       ((𝜓 → (𝜑𝜒)) ↔ (𝜑 ↔ (𝜓𝜒)))
 
TheorembiadaniALT 817 Alternate proof of biadani 816 not using biadan 815. (Contributed by BJ, 4-Mar-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       ((𝜓 → (𝜑𝜒)) ↔ (𝜑 ↔ (𝜓𝜒)))
 
Theorembiadanii 818 Inference associated with biadani 816. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.)
(𝜑𝜓)    &   (𝜓 → (𝜑𝜒))       (𝜑 ↔ (𝜓𝜒))
 
Theorempm5.1 819 Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
((𝜑𝜓) → (𝜑𝜓))
 
Theorempm5.21 820 Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.)
((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑𝜓))
 
Theorempm5.35 821 Theorem *5.35 of [WhiteheadRussell] p. 125. Closed form of 2thd 266. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
 
Theoremabai 822 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.)
((𝜑𝜓) ↔ (𝜑 ∧ (𝜑𝜓)))
 
Theorempm4.45im 823 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
(𝜑 ↔ (𝜑 ∧ (𝜓𝜑)))
 
Theoremimpimprbi 824 An implication and its reverse are equivalent exactly when both operands are equivalent. The right hand side resembles that of dfbi2 475, but is a weaker operator than . Note that an implication and its reverse can never be simultaneously false, because of pm2.521 177. (Contributed by Wolf Lammen, 18-Dec-2023.)
((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜓𝜑)))
 
Theoremnan 825 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))
 
Theorempm5.31 826 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜒 ∧ (𝜑𝜓)) → (𝜑 → (𝜓𝜒)))
 
Theorempm5.31r 827 Variant of pm5.31 826. (Contributed by Rodolfo Medina, 15-Oct-2010.)
((𝜒 ∧ (𝜑𝜓)) → (𝜑 → (𝜒𝜓)))
 
Theorempm4.15 828 Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
(((𝜑𝜓) → ¬ 𝜒) ↔ ((𝜓𝜒) → ¬ 𝜑))
 
Theorempm5.36 829 Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∧ (𝜑𝜓)) ↔ (𝜓 ∧ (𝜑𝜓)))
 
Theoremannotanannot 830 A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022.) (Proof shortened by Wolf Lammen, 1-Apr-2022.)
((𝜑 ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ∧ ¬ 𝜓))
 
Theorempm5.33 831 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ((𝜑𝜓) → 𝜒)))
 
Theoremsyl12anc 832 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       (𝜑𝜏)
 
Theoremsyl21anc 833 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (𝜑𝜏)
 
Theoremsyl22anc 834 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (((𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑𝜂)
 
Theoremsyl1111anc 835 Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1366 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((((𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)       (𝜑𝜂)
 
Theoremmpsyl4anc 836 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   𝜓    &   𝜒    &   (𝜃𝜏)    &   ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂)       (𝜃𝜂)
 
Theorempm4.87 837 Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
(((((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))) ∧ ((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))) ∧ ((𝜓 → (𝜑𝜒)) ↔ ((𝜓𝜑) → 𝜒)))
 
Theorembimsc1 838 Removal of conjunct from one side of an equivalence. (Contributed by NM, 21-Jun-1993.)
(((𝜑𝜓) ∧ (𝜒 ↔ (𝜓𝜑))) → (𝜒𝜑))
 
Theorema2and 839 Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
(𝜑 → ((𝜓𝜌) → (𝜏𝜃)))    &   (𝜑 → ((𝜓𝜌) → 𝜒))       (𝜑 → (((𝜓𝜒) → 𝜏) → ((𝜓𝜌) → 𝜃)))
 
Theoremanimpimp2impd 840 Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.)
((𝜓𝜑) → (𝜒 → (𝜃𝜂)))    &   ((𝜓 ∧ (𝜑𝜃)) → (𝜂𝜏))       (𝜑 → ((𝜓𝜒) → (𝜓 → (𝜃𝜏))))
 
1.2.7  Logical disjunction

This section defines disjunction of two formulas, denoted by infix " " and read "or". It is defined in terms of implication and negation, which is possible in classical logic (but not in intuitionistic logic: see iset.mm). This section contains only theorems proved without df-an 397 (theorems that are proved using df-an 397 are deferred to the next section). Basic theorems that help simplifying and applying disjunction are olc 862, orc 861, and orcom 864.

As mentioned in the "note on definitions" in the section comment for logical equivalence, all theorems in this and the previous section can be stated in terms of implication and negation only. Additionally, in classical logic (but not in intuitionistic logic: see iset.mm), it is also possible to translate conjunction into disjunction and conversely via the De Morgan law anor 976: conjunction and disjunction are dual connectives. Either is sufficient to develop all propositional calculus of the logic (together with implication and negation). In practice, conjunction is more efficient, its big advantage being the possibility to use it to group antecedents in a convenient way, using imp 407 and ex 413 as noted in the previous section.

An illustration of the conservativity of df-an 397 is given by orim12dALT 905, which is an alternate proof of orim12d 958 not using df-an 397.

 
Syntaxwo 841 Extend wff definition to include disjunction ("or").
wff (𝜑𝜓)
 
Definitiondf-or 842 Define disjunction (logical "or"). Definition of [Margaris] p. 49. When the left operand, right operand, or both are true, the result is true; when both sides are false, the result is false. For example, it is true that (2 = 3 ∨ 4 = 4) (ex-or 28128). After we define the constant true (df-tru 1531) and the constant false (df-fal 1541), we will be able to prove these truth table values: ((⊤ ∨ ⊤) ↔ ⊤) (truortru 1565), ((⊤ ∨ ⊥) ↔ ⊤) (truorfal 1566), ((⊥ ∨ ⊤) ↔ ⊤) (falortru 1567), and ((⊥ ∨ ⊥) ↔ ⊥) (falorfal 1568).

Contrast with (df-an 397), (wi 4), (df-nan 1476), and (df-xor 1496). (Contributed by NM, 27-Dec-1992.)

((𝜑𝜓) ↔ (¬ 𝜑𝜓))
 
Theorempm4.64 843 Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜓) ↔ (𝜑𝜓))
 
Theorempm4.66 844 Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))
 
Theorempm2.53 845 Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → (¬ 𝜑𝜓))
 
Theorempm2.54 846 Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜓) → (𝜑𝜓))
 
Theoremimor 847 Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.)
((𝜑𝜓) ↔ (¬ 𝜑𝜓))
 
Theoremimori 848 Infer disjunction from implication. (Contributed by NM, 12-Mar-2012.)
(𝜑𝜓)       𝜑𝜓)
 
Theoremimorri 849 Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝜑𝜓)       (𝜑𝜓)
 
Theorempm4.62 850 Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
 
Theoremjaoi 851 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜓)       ((𝜑𝜒) → 𝜓)
 
Theoremjao1i 852 Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)
(𝜓 → (𝜒𝜑))       ((𝜑𝜓) → (𝜒𝜑))
 
Theoremjaod 853 Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → ((𝜓𝜃) → 𝜒))
 
Theoremmpjaod 854 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)
 
Theoremori 855 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
(𝜑𝜓)       𝜑𝜓)
 
Theoremorri 856 Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.)
𝜑𝜓)       (𝜑𝜓)
 
Theoremorrd 857 Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.)
(𝜑 → (¬ 𝜓𝜒))       (𝜑 → (𝜓𝜒))
 
Theoremord 858 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓𝜒))
 
Theoremorci 859 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
𝜑       (𝜑𝜓)
 
Theoremolci 860 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
𝜑       (𝜓𝜑)
 
Theoremorc 861 Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.)
(𝜑 → (𝜑𝜓))
 
Theoremolc 862 Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)
(𝜑 → (𝜓𝜑))
 
Theorempm1.4 863 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremorcom 864 Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
Theoremorcomd 865 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))
 
Theoremunitresl 866 A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ¬ 𝜒)       (𝜑𝜓)
 
Theoremunitresr 867 A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)
 
Theoremorcoms 868 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremorcd 869 Deduction introducing a disjunct. A translation of natural deduction rule IR ( insertion right), see natded 28110. (Contributed by NM, 20-Sep-2007.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒))
 
Theoremolcd 870 Deduction introducing a disjunct. A translation of natural deduction rule IL ( insertion left), see natded 28110. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))
 
Theoremorcs 871 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 17) -type inference in a proof. (Contributed by NM, 21-Jun-1994.)
((𝜑𝜓) → 𝜒)       (𝜑𝜒)
 
Theoremolcs 872 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)
 
Theoremmtord 873 A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ¬ 𝜓)
 
Theorempm3.2ni 874 Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
¬ 𝜑    &    ¬ 𝜓        ¬ (𝜑𝜓)
 
Theorempm2.45 875 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜑)
 
Theorempm2.46 876 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜓)
 
Theorempm2.47 877 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑𝜓))
 
Theorempm2.48 878 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 ∨ ¬ 𝜓))
 
Theorempm2.49 879 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
 
Theoremnorbi 880 If neither of two propositions is true, then these propositions are equivalent. (Contributed by BJ, 26-Apr-2019.)
(¬ (𝜑𝜓) → (𝜑𝜓))
 
Theoremnbior 881 If two propositions are not equivalent, then at least one is true. (Contributed by BJ, 19-Apr-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
(¬ (𝜑𝜓) → (𝜑𝜓))
 
Theoremorel1 882 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))
 
Theorempm2.25 883 Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
(𝜑 ∨ ((𝜑𝜓) → 𝜓))
 
Theoremorel2 884 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))
 
Theorempm2.67-2 885 Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜒) → 𝜓) → (𝜑𝜓))
 
Theorempm2.67 886 Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) → 𝜓) → (𝜑𝜓))
 
Theoremcurryax 887 A non-intuitionistic positive statement, sometimes called a paradox of material implication. Sometimes called Curry's axiom. Similar to exmid 888 (obtained by substituting for 𝜓) but positive. For another non-intuitionistic positive statement, see peirce 203. (Contributed by BJ, 4-Apr-2021.)
(𝜑 ∨ (𝜑𝜓))
 
Theoremexmid 888 Law of excluded middle, also called the principle of tertium non datur. Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is either true or not true; there are no in-between values of truth. This is an essential distinction of our classical logic and is not a theorem of intuitionistic logic. In intuitionistic logic, if this statement is true for some 𝜑, then 𝜑 is decidable. (Contributed by NM, 29-Dec-1992.)
(𝜑 ∨ ¬ 𝜑)
 
Theoremexmidd 889 Law of excluded middle in a context. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑 → (𝜓 ∨ ¬ 𝜓))
 
Theorempm2.1 890 Theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
𝜑𝜑)
 
Theorempm2.13 891 Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(𝜑 ∨ ¬ ¬ ¬ 𝜑)
 
Theorempm2.621 892 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))
 
Theorempm2.62 893 Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))
 
Theorempm2.68 894 Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) → 𝜓) → (𝜑𝜓))
 
Theoremdfor2 895 Logical 'or' expressed in terms of implication only. Theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Wolf Lammen, 20-Oct-2012.)
((𝜑𝜓) ↔ ((𝜑𝜓) → 𝜓))
 
Theorempm2.07 896 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (𝜑𝜑))
 
Theorempm1.2 897 Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut". (Contributed by NM, 3-Jan-2005.)
((𝜑𝜑) → 𝜑)
 
Theoremoridm 898 Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.)
((𝜑𝜑) ↔ 𝜑)
 
Theorempm4.25 899 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑𝜑))
 
Theorempm2.4 900 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))
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