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	pm5.74da 801	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 802	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 803	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	pm4.14 804	Theorem *4.14 of [WhiteheadRussell] p. 117. Related to con34b 316. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))
Theorem	pm3.37 805	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 806	Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 609. 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	pm3.4 807	Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
⊢ ((𝜑 ∧ 𝜓) → (𝜑 → 𝜓))
Theorem	exbiri 808	Inference form of exbir 42098. This proof is exbiriVD 42474 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 809	Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	pm2.61dan 810	Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	pm2.61ddan 811	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 812	Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.)
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃)    &   ⊢ ((𝜑 ∧ ¬ 𝜒) → 𝜃)    &   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	mtand 813	A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm2.65da 814	Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	condan 815	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 816	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 819 and elelb 35082. (Contributed by BJ, 4-Mar-2023.) (Proof shortened by Wolf Lammen, 8-Mar-2023.)
⊢ ((𝜑 → 𝜓) ↔ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒))))
Theorem	biadani 817	Inference associated with biadan 816. (Contributed by BJ, 4-Mar-2023.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	biadaniALT 818	Alternate proof of biadani 817 not using biadan 816. (Contributed by BJ, 4-Mar-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	biadanii 819	Inference associated with biadani 817. 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 820	Deduction associated with biadani 817. 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	pm5.1 821	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	pm5.21 822	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	pm5.35 823	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 824	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 825	Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
⊢ (𝜑 ↔ (𝜑 ∧ (𝜓 → 𝜑)))
Theorem	impimprbi 826	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 pm2.521 176. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)))
Theorem	nan 827	Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒))
Theorem	pm5.31 828	Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜓 ∧ 𝜒)))
Theorem	pm5.31r 829	Variant of pm5.31 828. (Contributed by Rodolfo Medina, 15-Oct-2010.)
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜒 ∧ 𝜓)))
Theorem	pm4.15 830	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	pm5.36 831	Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜑 ↔ 𝜓)) ↔ (𝜓 ∧ (𝜑 ↔ 𝜓)))
Theorem	annotanannot 832	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	pm5.33 833	Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜓 → 𝜒)) ↔ (𝜑 ∧ ((𝜑 ∧ 𝜓) → 𝜒)))
Theorem	syl12anc 834	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 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	syl22anc 836	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 837	Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1373 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 838	Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	mpsyl4anc 839	An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ (𝜃 → 𝜏)    &   ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜃 → 𝜂)
Theorem	pm4.87 840	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 841	Removal of conjunct from one side of an equivalence. (Contributed by NM, 21-Jun-1993.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑))
Theorem	a2and 842	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 843	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 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 865, orc 864, and orcom 867. 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 407 and ex 413 as noted in the previous section. An illustration of the conservativity of df-an 397 is given by orim12dALT 909, which is an alternate proof of orim12d 962 not using df-an 397.
Syntax	wo 844	Extend wff definition to include disjunction ("or").
wff (𝜑 ∨ 𝜓)
Definition	df-or 845	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 28785). After we define the constant true ⊤ (df-tru 1542) and the constant false ⊥ (df-fal 1552), we will be able to prove these truth table values: ((⊤ ∨ ⊤) ↔ ⊤) (truortru 1576), ((⊤ ∨ ⊥) ↔ ⊤) (truorfal 1577), ((⊥ ∨ ⊤) ↔ ⊤) (falortru 1578), and ((⊥ ∨ ⊥) ↔ ⊥) (falorfal 1579). Contrast with ∧ (df-an 397), → (wi 4), ⊼ (df-nan 1487), and ⊻ (df-xor 1507). (Contributed by NM, 27-Dec-1992.)
⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
Theorem	pm4.64 846	Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓))
Theorem	pm4.66 847	Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))
Theorem	pm2.53 848	Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
Theorem	pm2.54 849	Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))
Theorem	imor 850	Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.)
⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
Theorem	imori 851	Infer disjunction from implication. (Contributed by NM, 12-Mar-2012.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (¬ 𝜑 ∨ 𝜓)
Theorem	imorri 852	Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (¬ 𝜑 ∨ 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm4.62 853	Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
Theorem	jaoi 854	Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜒) → 𝜓)
Theorem	jao1i 855	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 856	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 857	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 858	Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
⊢ (𝜑 ∨ 𝜓)    ⇒   ⊢ (¬ 𝜑 → 𝜓)
Theorem	orri 859	Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.)
⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ (𝜑 ∨ 𝜓)
Theorem	orrd 860	Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.)
⊢ (𝜑 → (¬ 𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ∨ 𝜒))
Theorem	ord 861	Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
Theorem	orci 862	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
⊢ 𝜑    ⇒   ⊢ (𝜑 ∨ 𝜓)
Theorem	olci 863	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
⊢ 𝜑    ⇒   ⊢ (𝜓 ∨ 𝜑)
Theorem	orc 864	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.)
⊢ (𝜑 → (𝜑 ∨ 𝜓))
Theorem	olc 865	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)
⊢ (𝜑 → (𝜓 ∨ 𝜑))
Theorem	pm1.4 866	Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → (𝜓 ∨ 𝜑))
Theorem	orcom 867	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 868	Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ∨ 𝜓))
Theorem	orcoms 869	Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
⊢ ((𝜑 ∨ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜓 ∨ 𝜑) → 𝜒)
Theorem	orcd 870	Deduction introducing a disjunct. A translation of natural deduction rule ∨ IR (∨ insertion right), see natded 28767. (Contributed by NM, 20-Sep-2007.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ∨ 𝜒))
Theorem	olcd 871	Deduction introducing a disjunct. A translation of natural deduction rule ∨ IL (∨ insertion left), see natded 28767. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 ∨ 𝜓))
Theorem	orcs 872	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 873	Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ ((𝜑 ∨ 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	olcnd 874	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	unitreslOLD 875	Obsolete version of olcnd 874 as of 13-Apr-2024. (Contributed by Giovanni Mascellani, 15-Sep-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (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))