P. 9 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 801-900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pm3.35 801	Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. Variant of pm2.27 42. (Contributed by NM, 14-Dec-2002.)
⊢ ((𝜑 ∧ (𝜑 → 𝜓)) → 𝜓)
Theorem	pm5.74da 802	Distribution of implication over biconditional (deduction form). Variant of pm5.74d 272. (Contributed by NM, 4-May-2007.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))
Theorem	bitr 803	Theorem *4.22 of [WhiteheadRussell] p. 117. bitri 274 in closed form. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜓 ↔ 𝜒)) → (𝜑 ↔ 𝜒))
Theorem	biantr 804	A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜓)) → (𝜑 ↔ 𝜒))
Theorem	pm4.14 805	Theorem *4.14 of [WhiteheadRussell] p. 117. Related to con34b 315. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))
Theorem	pm3.37 806	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))
Theorem	anim12 807	Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 607. 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.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))
Theorem	pm3.4 808	Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
⊢ ((𝜑 ∧ 𝜓) → (𝜑 → 𝜓))
Theorem	exbiri 809	Inference form of exbir 43948. This proof is exbiriVD 44324 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))
Theorem	pm2.61ian 810	Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	pm2.61dan 811	Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	pm2.61ddan 812	Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    &   ⊢ ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	pm2.61dda 813	Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.)
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃)    &   ⊢ ((𝜑 ∧ ¬ 𝜒) → 𝜃)    &   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	mtand 814	A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm2.65da 815	Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	condan 816	Proof by contradiction. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Wolf Lammen, 19-Jun-2014.)
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	biadan 817	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 820 and elelb 36408. (Contributed by BJ, 4-Mar-2023.) (Proof shortened by Wolf Lammen, 8-Mar-2023.)
⊢ ((𝜑 → 𝜓) ↔ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒))))
Theorem	biadani 818	Inference associated with biadan 817. (Contributed by BJ, 4-Mar-2023.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	biadaniALT 819	Alternate proof of biadani 818 not using biadan 817. (Contributed by BJ, 4-Mar-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	biadanii 820	Inference associated with biadani 818. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
Theorem	biadanid 821	Deduction associated with biadani 818. Add a conjunction to an equivalence. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
Theorem	pm5.1 822	Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓))
Theorem	pm5.21 823	Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.)
⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ 𝜓))
Theorem	pm5.35 824	Theorem *5.35 of [WhiteheadRussell] p. 125. Closed form of 2thd 264. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒)))
Theorem	abai 825	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.)
⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 → 𝜓)))
Theorem	pm4.45im 826	Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
⊢ (𝜑 ↔ (𝜑 ∧ (𝜓 → 𝜑)))
Theorem	impimprbi 827	An implication and its reverse are equivalent exactly when both operands are equivalent. The right hand side resembles that of dfbi2 473, but ↔ is a weaker operator than ∧. Note that an implication and its reverse can never be simultaneously false, because of pm2.521 176. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)))
Theorem	nan 828	Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒))
Theorem	pm5.31 829	Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜓 ∧ 𝜒)))
Theorem	pm5.31r 830	Variant of pm5.31 829. (Contributed by Rodolfo Medina, 15-Oct-2010.)
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜒 ∧ 𝜓)))
Theorem	pm4.15 831	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))
Theorem	pm5.36 832	Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜑 ↔ 𝜓)) ↔ (𝜓 ∧ (𝜑 ↔ 𝜓)))
Theorem	annotanannot 833	A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022.) (Proof shortened by Wolf Lammen, 1-Apr-2022.)
⊢ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ ¬ 𝜓))
Theorem	pm5.33 834	Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜓 → 𝜒)) ↔ (𝜑 ∧ ((𝜑 ∧ 𝜓) → 𝜒)))
Theorem	syl12anc 835	Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	syl21anc 836	Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	syl22anc 837	Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜂)    ⇒   ⊢ (𝜑 → 𝜂)
Theorem	syl1111anc 838	Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1371 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ ((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → 𝜂)
Theorem	syldbl2 839	Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	mpsyl4anc 840	An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ (𝜃 → 𝜏)    &   ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜃 → 𝜂)
Theorem	pm4.87 841	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 → (𝜓 → 𝜒))) ∧ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜓 → (𝜑 → 𝜒)))) ∧ ((𝜓 → (𝜑 → 𝜒)) ↔ ((𝜓 ∧ 𝜑) → 𝜒)))
Theorem	bimsc1 842	Removal of conjunct from one side of an equivalence. (Contributed by NM, 21-Jun-1993.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑))
Theorem	a2and 843	Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
⊢ (𝜑 → ((𝜓 ∧ 𝜌) → (𝜏 → 𝜃)))    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → 𝜒))    ⇒   ⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜏) → ((𝜓 ∧ 𝜌) → 𝜃)))
Theorem	animpimp2impd 844	Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.)
⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜂)))    &   ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → (𝜂 → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏))))
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 395 (theorems that are proved using df-an 395 are deferred to the next section). Basic theorems that help simplifying and applying disjunction are olc 866, orc 865, and orcom 868. 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 980: 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 405 and ex 411 as noted in the previous section. An illustration of the conservativity of df-an 395 is given by orim12dALT 909, which is an alternate proof of orim12d 962 not using df-an 395.
Syntax	wo 845	Extend wff definition to include disjunction ("or").
wff (𝜑 ∨ 𝜓)
Definition	df-or 846	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 30251). After we define the constant true ⊤ (df-tru 1536) and the constant false ⊥ (df-fal 1546), we will be able to prove these truth table values: ((⊤ ∨ ⊤) ↔ ⊤) (truortru 1570), ((⊤ ∨ ⊥) ↔ ⊤) (truorfal 1571), ((⊥ ∨ ⊤) ↔ ⊤) (falortru 1572), and ((⊥ ∨ ⊥) ↔ ⊥) (falorfal 1573). Contrast with ∧ (df-an 395), → (wi 4), ⊼ (df-nan 1485), and ⊻ (df-xor 1505). (Contributed by NM, 27-Dec-1992.)
⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
Theorem	pm4.64 847	Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓))
Theorem	pm4.66 848	Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))
Theorem	pm2.53 849	Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
Theorem	pm2.54 850	Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))
Theorem	imor 851	Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.)
⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
Theorem	imori 852	Infer disjunction from implication. (Contributed by NM, 12-Mar-2012.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (¬ 𝜑 ∨ 𝜓)
Theorem	imorri 853	Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (¬ 𝜑 ∨ 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm4.62 854	Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
Theorem	jaoi 855	Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜒) → 𝜓)
Theorem	jao1i 856	Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)
⊢ (𝜓 → (𝜒 → 𝜑))    ⇒   ⊢ ((𝜑 ∨ 𝜓) → (𝜒 → 𝜑))
Theorem	jaod 857	Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 → 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → 𝜒))
Theorem	mpjaod 858	Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 → 𝜒))    &   ⊢ (𝜑 → (𝜓 ∨ 𝜃))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	ori 859	Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
⊢ (𝜑 ∨ 𝜓)    ⇒   ⊢ (¬ 𝜑 → 𝜓)
Theorem	orri 860	Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.)
⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ (𝜑 ∨ 𝜓)
Theorem	orrd 861	Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.)
⊢ (𝜑 → (¬ 𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ∨ 𝜒))
Theorem	ord 862	Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
Theorem	orci 863	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
⊢ 𝜑    ⇒   ⊢ (𝜑 ∨ 𝜓)
Theorem	olci 864	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
⊢ 𝜑    ⇒   ⊢ (𝜓 ∨ 𝜑)
Theorem	orc 865	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.)
⊢ (𝜑 → (𝜑 ∨ 𝜓))
Theorem	olc 866	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)
⊢ (𝜑 → (𝜓 ∨ 𝜑))
Theorem	pm1.4 867	Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → (𝜓 ∨ 𝜑))
Theorem	orcom 868	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.)
⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
Theorem	orcomd 869	Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ∨ 𝜓))
Theorem	orcoms 870	Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
⊢ ((𝜑 ∨ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜓 ∨ 𝜑) → 𝜒)
Theorem	orcd 871	Deduction introducing a disjunct. A translation of natural deduction rule ∨ IR (∨ insertion right), see natded 30233. (Contributed by NM, 20-Sep-2007.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ∨ 𝜒))
Theorem	olcd 872	Deduction introducing a disjunct. A translation of natural deduction rule ∨ IL (∨ insertion left), see natded 30233. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 ∨ 𝜓))
Theorem	orcs 873	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.)
⊢ ((𝜑 ∨ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	olcs 874	Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ ((𝜑 ∨ 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	olcnd 875	A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2024.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    &   ⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	orcnd 876	A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	mtord 877	A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → ¬ 𝜃)    &   ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm3.2ni 878	Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
⊢ ¬ 𝜑    &   ⊢ ¬ 𝜓    ⇒   ⊢ ¬ (𝜑 ∨ 𝜓)
Theorem	pm2.45 879	Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑)
Theorem	pm2.46 880	Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓)
Theorem	pm2.47 881	Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∨ 𝜓))
Theorem	pm2.48 882	Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ∨ ¬ 𝜓))
Theorem	pm2.49 883	Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
Theorem	norbi 884	If neither of two propositions is true, then these propositions are equivalent. (Contributed by BJ, 26-Apr-2019.)
⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓))
Theorem	nbior 885	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.)
⊢ (¬ (𝜑 ↔ 𝜓) → (𝜑 ∨ 𝜓))
Theorem	orel1 886	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))
Theorem	pm2.25 887	Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 ∨ ((𝜑 ∨ 𝜓) → 𝜓))
Theorem	orel2 888	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))
Theorem	pm2.67-2 889	Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜑 → 𝜓))
Theorem	pm2.67 890	Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ∨ 𝜓) → 𝜓) → (𝜑 → 𝜓))
Theorem	curryax 891	A non-intuitionistic positive statement, sometimes called a paradox of material implication. Sometimes called Curry's axiom. Similar to exmid 892 (obtained by substituting ⊥ for 𝜓) but positive. For another non-intuitionistic positive statement, see peirce 201. (Contributed by BJ, 4-Apr-2021.)
⊢ (𝜑 ∨ (𝜑 → 𝜓))
Theorem	exmid 892	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)
Theorem	exmidd 893	Law of excluded middle in a context. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓))
Theorem	pm2.1 894	Theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
⊢ (¬ 𝜑 ∨ 𝜑)
Theorem	pm2.13 895	Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 ∨ ¬ ¬ ¬ 𝜑)
Theorem	pm2.621 896	Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → ((𝜑 ∨ 𝜓) → 𝜓))
Theorem	pm2.62 897	Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
⊢ ((𝜑 ∨ 𝜓) → ((𝜑 → 𝜓) → 𝜓))
Theorem	pm2.68 898	Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓))
Theorem	dfor2 899	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.)
⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 → 𝜓) → 𝜓))
Theorem	pm2.07 900	Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 → (𝜑 ∨ 𝜑))