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	simp-10r 801	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
⊢ (((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)
Theorem	simp-11l 802	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
⊢ ((((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜑)
Theorem	simp-11r 803	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 24-May-2022.)
⊢ ((((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)
Theorem	pm2.01da 804	Deduction based on reductio ad absurdum. See pm2.01 189. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm2.18da 805	Deduction based on reductio ad absurdum. See pm2.18 128. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	impbida 806	Deduce an equivalence from two implications. Variant of impbid 213. (Contributed by NM, 17-Feb-2007.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm5.21nd 807	Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd 380. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    &   ⊢ (𝜃 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm3.35 808	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 809	Distribution of implication over biconditional (deduction form). Variant of pm5.74d 274. (Contributed by NM, 4-May-2007.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))
Theorem	bitr 810	Theorem *4.22 of [WhiteheadRussell] p. 117. bitri 276 in closed form. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜓 ↔ 𝜒)) → (𝜑 ↔ 𝜒))
Theorem	biantr 811	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	pm4.14 812	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.)
⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))
Theorem	pm3.37 813	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 814	Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 615. 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	pm3.4 815	Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
⊢ ((𝜑 ∧ 𝜓) → (𝜑 → 𝜓))
Theorem	exbiri 816	Inference form of exbir 44930. This proof is exbiriVD 45304 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 817	Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	pm2.61dan 818	Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	pm2.61ddan 819	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 820	Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.)
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃)    &   ⊢ ((𝜑 ∧ ¬ 𝜒) → 𝜃)    &   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	mtand 821	A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm2.65da 822	Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	condan 823	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 824	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 827 and elelb 37257. (Contributed by BJ, 4-Mar-2023.) (Proof shortened by Wolf Lammen, 8-Mar-2023.)
⊢ ((𝜑 → 𝜓) ↔ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒))))
Theorem	biadani 825	Inference associated with biadan 824. (Contributed by BJ, 4-Mar-2023.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	biadaniALT 826	Alternate proof of biadani 825 not using biadan 824. (Contributed by BJ, 4-Mar-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	biadanii 827	Inference associated with biadani 825. 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 828	Deduction associated with biadani 825. 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	pm5.1 829	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	pm5.21 830	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	pm5.35 831	Theorem *5.35 of [WhiteheadRussell] p. 125. Closed form of 2thd 266. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒)))
Theorem	abai 832	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 833	Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
⊢ (𝜑 ↔ (𝜑 ∧ (𝜓 → 𝜑)))
Theorem	impimprbi 834	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 835	Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒))
Theorem	pm5.31 836	Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜓 ∧ 𝜒)))
Theorem	pm5.31r 837	Variant of pm5.31 836. (Contributed by Rodolfo Medina, 15-Oct-2010.)
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜒 ∧ 𝜓)))
Theorem	pm4.15 838	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	pm5.36 839	Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜑 ↔ 𝜓)) ↔ (𝜓 ∧ (𝜑 ↔ 𝜓)))
Theorem	annotanannot 840	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	pm5.33 841	Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜓 → 𝜒)) ↔ (𝜑 ∧ ((𝜑 ∧ 𝜓) → 𝜒)))
Theorem	syl12anc 842	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 843	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 844	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	bibiad 845	Eliminate an hypothesis 𝜃 in a biconditional. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	syl1111anc 846	Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1382 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 847	Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	mpsyl4anc 848	An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ (𝜃 → 𝜏)    &   ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜃 → 𝜂)
Theorem	pm4.87 849	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 850	Removal of conjunct from one side of an equivalence. (Contributed by NM, 21-Jun-1993.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑))
Theorem	a2and 851	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 852	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 874, orc 873, and orcom 876. 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 990: 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 917, which is an alternate proof of orim12d 972 not using df-an 397.
Syntax	wo 853	Extend wff definition to include disjunction ("or").
wff (𝜑 ∨ 𝜓)
Definition	df-or 854	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 30516). After we define the constant true ⊤ (df-tru 1550) and the constant false ⊥ (df-fal 1560), we will be able to prove these truth table values: ((⊤ ∨ ⊤) ↔ ⊤) (truortru 1584), ((⊤ ∨ ⊥) ↔ ⊤) (truorfal 1585), ((⊥ ∨ ⊤) ↔ ⊤) (falortru 1586), and ((⊥ ∨ ⊥) ↔ ⊥) (falorfal 1587). Contrast with ∧ (df-an 397), → (wi 4), ⊼ (df-nan 1499), and ⊻ (df-xor 1519). (Contributed by NM, 27-Dec-1992.)
⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
Theorem	pm4.64 855	Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓))
Theorem	pm4.66 856	Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))
Theorem	pm2.53 857	Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
Theorem	pm2.54 858	Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))
Theorem	imor 859	Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.)
⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
Theorem	imori 860	Infer disjunction from implication. (Contributed by NM, 12-Mar-2012.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (¬ 𝜑 ∨ 𝜓)
Theorem	imorri 861	Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (¬ 𝜑 ∨ 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm4.62 862	Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
Theorem	jaoi 863	Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜒) → 𝜓)
Theorem	jao1i 864	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 865	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 866	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 867	Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
⊢ (𝜑 ∨ 𝜓)    ⇒   ⊢ (¬ 𝜑 → 𝜓)
Theorem	orri 868	Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.)
⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ (𝜑 ∨ 𝜓)
Theorem	orrd 869	Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.)
⊢ (𝜑 → (¬ 𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ∨ 𝜒))
Theorem	ord 870	Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
Theorem	orci 871	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
⊢ 𝜑    ⇒   ⊢ (𝜑 ∨ 𝜓)
Theorem	olci 872	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
⊢ 𝜑    ⇒   ⊢ (𝜓 ∨ 𝜑)
Theorem	orc 873	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.)
⊢ (𝜑 → (𝜑 ∨ 𝜓))
Theorem	olc 874	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)
⊢ (𝜑 → (𝜓 ∨ 𝜑))
Theorem	pm1.4 875	Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → (𝜓 ∨ 𝜑))
Theorem	orcom 876	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 877	Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ∨ 𝜓))
Theorem	orcoms 878	Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
⊢ ((𝜑 ∨ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜓 ∨ 𝜑) → 𝜒)
Theorem	orcd 879	Deduction introducing a disjunct. A translation of natural deduction rule ∨ IR (∨ insertion right), see natded 30498. (Contributed by NM, 20-Sep-2007.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ∨ 𝜒))
Theorem	olcd 880	Deduction introducing a disjunct. A translation of natural deduction rule ∨ IL (∨ insertion left), see natded 30498. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 ∨ 𝜓))
Theorem	orcs 881	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 882	Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ ((𝜑 ∨ 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	olcnd 883	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 884	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 885	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 886	Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
⊢ ¬ 𝜑    &   ⊢ ¬ 𝜓    ⇒   ⊢ ¬ (𝜑 ∨ 𝜓)
Theorem	pm2.45 887	Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑)
Theorem	pm2.46 888	Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓)
Theorem	pm2.47 889	Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∨ 𝜓))
Theorem	pm2.48 890	Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ∨ ¬ 𝜓))
Theorem	pm2.49 891	Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
Theorem	norbi 892	If neither of two propositions is true, then these propositions are equivalent. (Contributed by BJ, 26-Apr-2019.)
⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓))
Theorem	nbior 893	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 894	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	pm2.25 895	Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 ∨ ((𝜑 ∨ 𝜓) → 𝜓))
Theorem	orel2 896	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	pm2.67-2 897	Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜑 → 𝜓))
Theorem	pm2.67 898	Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ∨ 𝜓) → 𝜓) → (𝜑 → 𝜓))
Theorem	curryax 899	A non-intuitionistic positive statement, sometimes called a paradox of material implication. Sometimes called Curry's axiom. Similar to exmid 900 (obtained by substituting ⊥ for 𝜓) but positive. For another non-intuitionistic positive statement, see peirce 203. (Contributed by BJ, 4-Apr-2021.)
⊢ (𝜑 ∨ (𝜑 → 𝜓))
Theorem	exmid 900	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)