P. 10 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 901-1000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	olc 901	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)
⊢ (𝜑 → (𝜓 ∨ 𝜑))
Theorem	pm1.4 902	Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → (𝜓 ∨ 𝜑))
Theorem	orcom 903	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 904	Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ∨ 𝜓))
Theorem	orcoms 905	Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
⊢ ((𝜑 ∨ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜓 ∨ 𝜑) → 𝜒)
Theorem	orcd 906	Deduction introducing a disjunct. A translation of natural deduction rule ∨ IR (∨ insertion right), see natded 27817. (Contributed by NM, 20-Sep-2007.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ∨ 𝜒))
Theorem	olcd 907	Deduction introducing a disjunct. A translation of natural deduction rule ∨ IL (∨ insertion left), see natded 27817. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 ∨ 𝜓))
Theorem	orcs 908	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 909	Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ ((𝜑 ∨ 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	mtord 910	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 911	Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
⊢ ¬ 𝜑    &   ⊢ ¬ 𝜓    ⇒   ⊢ ¬ (𝜑 ∨ 𝜓)
Theorem	pm2.45 912	Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑)
Theorem	pm2.46 913	Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓)
Theorem	pm2.47 914	Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∨ 𝜓))
Theorem	pm2.48 915	Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ∨ ¬ 𝜓))
Theorem	pm2.49 916	Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
Theorem	norbi 917	If neither of two propositions is true, then these propositions are equivalent. (Contributed by BJ, 26-Apr-2019.)
⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓))
Theorem	nbior 918	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 919	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	pm2.25 920	Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 ∨ ((𝜑 ∨ 𝜓) → 𝜓))
Theorem	orel2 921	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	pm2.67-2 922	Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜑 → 𝜓))
Theorem	pm2.67 923	Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ∨ 𝜓) → 𝜓) → (𝜑 → 𝜓))
Theorem	curryax 924	A non-intuitionistic positive statement, sometimes called a paradox of material implication. Sometimes called Curry's axiom. Similar to exmid 925 but positive. For another non-intuitionistic positive statement, see peirce 194. (Contributed by BJ, 4-Apr-2021.)
⊢ (𝜑 ∨ (𝜑 → 𝜓))
Theorem	exmid 925	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 926	Law of excluded middle in a context. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓))
Theorem	pm2.1 927	Theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
⊢ (¬ 𝜑 ∨ 𝜑)
Theorem	pm2.13 928	Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 ∨ ¬ ¬ ¬ 𝜑)
Theorem	pm2.621 929	Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → ((𝜑 ∨ 𝜓) → 𝜓))
Theorem	pm2.62 930	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	pm2.68 931	Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓))
Theorem	dfor2 932	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	pm2.07 933	Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 → (𝜑 ∨ 𝜑))
Theorem	pm1.2 934	Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut". (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜑) → 𝜑)
Theorem	oridm 935	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.)
⊢ ((𝜑 ∨ 𝜑) ↔ 𝜑)
Theorem	pm4.25 936	Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 ↔ (𝜑 ∨ 𝜑))
Theorem	pm2.4 937	Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ (𝜑 ∨ 𝜓)) → (𝜑 ∨ 𝜓))
Theorem	pm2.41 938	Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜓 ∨ (𝜑 ∨ 𝜓)) → (𝜑 ∨ 𝜓))
Theorem	orim12i 939	Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃))
Theorem	orim1i 940	Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜒))
Theorem	orim2i 941	Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓))
Theorem	orim12dALT 942	Alternate proof of orim12d 994 which does not depend on df-an 387. This is an illustration of the conservativity of definitions (definitions do not permit to prove additional theorems whose statements do not contain the defined symbol). (Contributed by Wolf Lammen, 8-Aug-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏)))
Theorem	orbi2i 943	Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))
Theorem	orbi1i 944	Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒))
Theorem	orbi12i 945	Infer the disjunction of two equivalences. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃))
Theorem	orbi2d 946	Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒)))
Theorem	orbi1d 947	Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜃)))
Theorem	orbi1 948	Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)))
Theorem	orbi12d 949	Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜏)))
Theorem	pm1.5 950	Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → (𝜓 ∨ (𝜑 ∨ 𝜒)))
Theorem	or12 951	Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) ↔ (𝜓 ∨ (𝜑 ∨ 𝜒)))
Theorem	orass 952	Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))
Theorem	pm2.31 953	Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜓) ∨ 𝜒))
Theorem	pm2.32 954	Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → (𝜑 ∨ (𝜓 ∨ 𝜒)))
Theorem	pm2.3 955	Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → (𝜑 ∨ (𝜒 ∨ 𝜓)))
Theorem	or32 956	A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∨ 𝜓))
Theorem	or4 957	Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)))
Theorem	or42 958	Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜃 ∨ 𝜓)))
Theorem	orordi 959	Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒)))
Theorem	orordir 960	Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜒)))
Theorem	orimdi 961	Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013.)
⊢ ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
Theorem	pm2.76 962	Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ (𝜓 → 𝜒)) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
Theorem	pm2.85 963	Theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒)))
Theorem	pm2.75 964	Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.)
⊢ ((𝜑 ∨ 𝜓) → ((𝜑 ∨ (𝜓 → 𝜒)) → (𝜑 ∨ 𝜒)))
Theorem	pm4.78 965	Implication distributes over disjunction. Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒)))
Theorem	biort 966	A wff disjoined with truth is true. (Contributed by NM, 23-May-1999.)
⊢ (𝜑 → (𝜑 ↔ (𝜑 ∨ 𝜓)))
Theorem	biorf 967	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 ↔ (𝜑 ∨ 𝜓)))
Theorem	biortn 968	A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
⊢ (𝜑 → (𝜓 ↔ (¬ 𝜑 ∨ 𝜓)))
Theorem	biorfi 969	A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 16-Jul-2021.)
⊢ ¬ 𝜑    ⇒   ⊢ (𝜓 ↔ (𝜓 ∨ 𝜑))
Theorem	pm2.26 970	Theorem *2.26 of [WhiteheadRussell] p. 104. See pm2.27 42. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
⊢ (¬ 𝜑 ∨ ((𝜑 → 𝜓) → 𝜓))
Theorem	pm2.63 971	Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → ((¬ 𝜑 ∨ 𝜓) → 𝜓))
Theorem	pm2.64 972	Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑))
Theorem	pm2.42 973	Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
⊢ ((¬ 𝜑 ∨ (𝜑 → 𝜓)) → (𝜑 → 𝜓))
Theorem	pm5.11 974	Theorem *5.11 of [WhiteheadRussell] p. 123. See pm2.5 166. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓))
Theorem	pm5.12 975	Theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓))
Theorem	pm5.14 976	Theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) ∨ (𝜓 → 𝜒))
Theorem	pm5.13 977	Theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
⊢ ((𝜑 → 𝜓) ∨ (𝜓 → 𝜑))
Theorem	pm5.55 978	Theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 20-Jan-2013.)
⊢ (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓))
Theorem	pm4.72 979	Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.)
⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))
Theorem	imimorb 980	Simplify an implication between implications. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
⊢ (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒)))
Theorem	oibabs 981	Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) ↔ (𝜑 ↔ 𝜓))
Theorem	orbidi 982	Disjunction distributes over the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384. (Contributed by NM, 8-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Feb-2013.)
⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ 𝜒)))
Theorem	pm5.7 983	Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 982. (Contributed by Roy F. Longton, 21-Jun-2005.)
⊢ (((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) ↔ (𝜒 ∨ (𝜑 ↔ 𝜓)))
1.2.8  Mixed connectives This section gathers theorems of propositional calculus which use (either in their statement or proof) mixed connectives (at least conjunction and disjunction). As noted in the "note on definitions" in the section comment for logical equivalence, some theorem statements may contain for instance only conjunction or only disjunction, but both definitions are used in their proofs to make them shorter (this is exemplified in orim12d 994 versus orim12dALT 942). These theorems are mostly grouped at the beginning of this section. The family of theorems starting with animorl 1007 focus on the relation between conjunction and disjunction and can be seen as the starting point of mixed connectives in statements. This sectioning is not rigorously true, since for instance the section begins with jaao 984 and related theorems.
Theorem	jaao 984	Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → (𝜏 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∨ 𝜏) → 𝜒))
Theorem	jaoa 985	Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → (𝜏 → 𝜒))    ⇒   ⊢ ((𝜑 ∨ 𝜃) → ((𝜓 ∧ 𝜏) → 𝜒))
Theorem	jaoian 986	Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜃 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒)
Theorem	jaodan 987	Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃)) → 𝜒)
Theorem	mpjaodan 988	Eliminate a disjunction in a deduction. A translation of natural deduction rule ∨ E (∨ elimination), see natded 27817. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜒)    &   ⊢ (𝜑 → (𝜓 ∨ 𝜃))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	pm3.44 989	Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ (((𝜓 → 𝜑) ∧ (𝜒 → 𝜑)) → ((𝜓 ∨ 𝜒) → 𝜑))
Theorem	jao 990	Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 4-Apr-2013.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((𝜑 ∨ 𝜒) → 𝜓)))
Theorem	jaob 991	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))
Theorem	pm4.77 992	Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜓 → 𝜑) ∧ (𝜒 → 𝜑)) ↔ ((𝜓 ∨ 𝜒) → 𝜑))
Theorem	pm3.48 993	Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃)))
Theorem	orim12d 994	Disjoin antecedents and consequents in a deduction. See orim12dALT 942 for a proof which does not depend on df-an 387. (Contributed by NM, 10-May-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏)))
Theorem	orim1d 995	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜃)))
Theorem	orim2d 996	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ∨ 𝜓) → (𝜃 ∨ 𝜒)))
Theorem	orim2 997	Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜓 → 𝜒) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
Theorem	pm2.38 998	Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
⊢ ((𝜓 → 𝜒) → ((𝜓 ∨ 𝜑) → (𝜒 ∨ 𝜑)))
Theorem	pm2.36 999	Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
⊢ ((𝜓 → 𝜒) → ((𝜑 ∨ 𝜓) → (𝜒 ∨ 𝜑)))
Theorem	pm2.37 1000	Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
⊢ ((𝜓 → 𝜒) → ((𝜓 ∨ 𝜑) → (𝜑 ∨ 𝜒)))