P. 45 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4401-4500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pssnel 4401*	A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 → ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
Theorem	disjdif 4402	A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
Theorem	disjdifr 4403	A class and its relative complement are disjoint. (Contributed by Thierry Arnoux, 29-Nov-2023.)
⊢ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅
Theorem	difin0 4404	The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅
Theorem	unvdif 4405	The union of a class and its complement is the universe. Theorem 5.1(5) of [Stoll] p. 17. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∪ (V ∖ 𝐴)) = V
Theorem	undif1 4406	Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 4402). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
Theorem	undif2 4407	Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 4402). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)
⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
Theorem	undifabs 4408	Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴
Theorem	inundif 4409	The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴
Theorem	disjdif2 4410	The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)
Theorem	difun2 4411	Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
Theorem	undif 4412	Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
Theorem	ssdifin0 4413	A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅)
Theorem	ssdifeq0 4414	A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
Theorem	ssundif 4415	A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)
⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶)
Theorem	difcom 4416	Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)
⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)
Theorem	pssdifcom1 4417	Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝐶 ∖ 𝐴) ⊊ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊊ 𝐴))
Theorem	pssdifcom2 4418	Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊊ (𝐶 ∖ 𝐵)))
Theorem	difdifdir 4419	Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)
⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))
Theorem	uneqdifeq 4420	Two ways to say that 𝐴 and 𝐵 partition 𝐶 (when 𝐴 and 𝐵 don't overlap and 𝐴 is a part of 𝐶). (Contributed by FL, 17-Nov-2008.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 ↔ (𝐶 ∖ 𝐴) = 𝐵))
Theorem	raldifeq 4421*	Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	r19.2z 4422*	Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1981). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
Theorem	r19.2zb 4423*	A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4422. Interestingly enough, 𝜑 does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)
⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))
Theorem	r19.3rz 4424*	Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	r19.28z 4425*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.3rzv 4426*	Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	r19.9rzv 4427*	Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	r19.28zv 4428*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.)
⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.37zv 4429*	Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.)
⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.45zv 4430*	Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.44zv 4431*	Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)))
Theorem	r19.27z 4432*	Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))
Theorem	r19.27zv 4433*	Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.)
⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))
Theorem	r19.36zv 4434*	Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003.)
⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)))
Theorem	ralidmw 4435*	Idempotent law for restricted quantifier. Weak version of ralidm 4439, which does not require ax-10 2139, ax-12 2173, but requires ax-8 2110. (Contributed by Gino Giotto, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rzal 4436*	Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 2-Sep-2024.)
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rzalALT 4437*	Alternate proof of rzal 4436. Shorter, but requiring df-clel 2817, ax-8 2110. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexn0 4438*	Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 2-Sep-2024.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
Theorem	ralidm 4439	Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) Reduce axiom usage. (Revised by Gino Giotto, 2-Sep-2024.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	ral0 4440	Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 2-Sep-2024.)
⊢ ∀𝑥 ∈ ∅ 𝜑
Theorem	ralf0 4441*	The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 2-Sep-2024.)
⊢ ¬ 𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)
Theorem	rexn0OLD 4442*	Obsolete version of rexn0 4438 as of 2-Sep-2024. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
Theorem	ralidmOLD 4443*	Obsolete version of ralidm 4439 as of 2-Sep-2024. (Contributed by NM, 28-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	ral0OLD 4444	Obsolete version of ral0 4440 as of 2-Sep-2024. (Contributed by NM, 20-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥 ∈ ∅ 𝜑
Theorem	ralf0OLD 4445*	Obsolete version of ralf0 4441 as of 2-Sep-2024. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)
Theorem	ralnralall 4446*	A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓))
Theorem	falseral0 4447*	A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.)
⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)
Theorem	raaan 4448*	Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
Theorem	raaanv 4449*	Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
Theorem	sbss 4450*	Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
⊢ ([𝑦 / 𝑥]𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴)
Theorem	sbcssg 4451	Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ⊆ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ⊆ ⦋𝐴 / 𝑥⦌𝐶))
Theorem	raaan2 4452*	Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4448. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
Theorem	2reu4lem 4453*	Lemma for 2reu4 4454. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
Theorem	2reu4 4454*	Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2656. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
Theorem	csbdif 4455	Distribution of class substitution over difference of two classes. (Contributed by ML, 14-Jul-2020.)
⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶)
2.1.15  The conditional operator for classes This subsection introduces the conditional operator for classes, denoted by if(𝜑, 𝐴, 𝐵) (see df-if 4457). It is the analogue for classes of the conditional operator for propositions, denoted by if-(𝜑, 𝜓, 𝜒) (see df-ifp 1060).
Syntax	cif 4456	Extend class notation to include the conditional operator for classes.
class if(𝜑, 𝐴, 𝐵)
Definition	df-if 4457*	Definition of the conditional operator for classes. The expression if(𝜑, 𝐴, 𝐵) is read "if 𝜑 then 𝐴 else 𝐵". See iftrue 4462 and iffalse 4465 for its values. In the mathematical literature, this operator is rarely defined formally but is implicit in informal definitions such as "let f(x)=0 if x=0 and 1/x otherwise". An important use for us is in conjunction with the weak deduction theorem, which is described in the next section, beginning at dedth 4514. (Contributed by NM, 15-May-1999.)
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
Theorem	dfif2 4458*	An alternate definition of the conditional operator df-if 4457 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))}
Theorem	dfif6 4459*	An alternate definition of the conditional operator df-if 4457 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
Theorem	ifeq1 4460	Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))
Theorem	ifeq2 4461	Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵))
Theorem	iftrue 4462	Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
Theorem	iftruei 4463	Inference associated with iftrue 4462. (Contributed by BJ, 7-Oct-2018.)
⊢ 𝜑    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴
Theorem	iftrued 4464	Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴)
Theorem	iffalse 4465	Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
Theorem	iffalsei 4466	Inference associated with iffalse 4465. (Contributed by BJ, 7-Oct-2018.)
⊢ ¬ 𝜑    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵
Theorem	iffalsed 4467	Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐵)
Theorem	ifnefalse 4468	When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse 4465 directly in this case. It happens, e.g., in oevn0 8307. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 ≠ 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
Theorem	ifsb 4469	Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ 𝐶 = if(𝜑, 𝐷, 𝐸)
Theorem	dfif3 4470*	Alternate definition of the conditional operator df-if 4457. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ 𝐶 = {𝑥 ∣ 𝜑}    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
Theorem	dfif4 4471*	Alternate definition of the conditional operator df-if 4457. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
⊢ 𝐶 = {𝑥 ∣ 𝜑}    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶)))
Theorem	dfif5 4472*	Alternate definition of the conditional operator df-if 4457. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false (see also ab0orv 4309). (Contributed by Gérard Lang, 18-Aug-2013.)
⊢ 𝐶 = {𝑥 ∣ 𝜑}    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐵) ∪ (((𝐴 ∖ 𝐵) ∩ 𝐶) ∪ ((𝐵 ∖ 𝐴) ∩ (V ∖ 𝐶))))
Theorem	ifssun 4473	A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)
Theorem	ifeq12 4474	Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))
Theorem	ifeq1d 4475	Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
Theorem	ifeq2d 4476	Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
Theorem	ifeq12d 4477	Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))
Theorem	ifbi 4478	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Theorem	ifbid 4479	Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))
Theorem	ifbieq1d 4480	Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))
Theorem	ifbieq2i 4481	Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
Theorem	ifbieq2d 4482	Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))
Theorem	ifbieq12i 4483	Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)
Theorem	ifbieq12d 4484	Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
Theorem	nfifd 4485	Deduction form of nfif 4486. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵))
Theorem	nfif 4486	Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵)
Theorem	ifeq1da 4487	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
Theorem	ifeq2da 4488	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
Theorem	ifeq12da 4489	Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
Theorem	ifbieq12d2 4490	Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
Theorem	ifclda 4491	Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
Theorem	ifeqda 4492	Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)
Theorem	elimif 4493	Elimination of a conditional operator contained in a wff 𝜓. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)
⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃)))
Theorem	ifbothda 4494	A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))    &   ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))    &   ⊢ ((𝜂 ∧ 𝜑) → 𝜓)    &   ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒)    ⇒   ⊢ (𝜂 → 𝜃)
Theorem	ifboth 4495	A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))    &   ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
Theorem	ifid 4496	Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
⊢ if(𝜑, 𝐴, 𝐴) = 𝐴
Theorem	eqif 4497	Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶)))
Theorem	ifval 4498	Another expression of the value of the if predicate, analogous to eqif 4497. See also the more specialized iftrue 4462 and iffalse 4465. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶)))
Theorem	elif 4499	Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)))
Theorem	ifel 4500	Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
⊢ (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑 ∧ 𝐴 ∈ 𝐶) ∨ (¬ 𝜑 ∧ 𝐵 ∈ 𝐶)))