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	disj 4401*	Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) Avoid ax-10 2146, ax-11 2162, ax-12 2182. (Revised by GG, 28-Jun-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
Theorem	disjr 4402*	Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ 𝐴)
Theorem	disj1 4403*	Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
Theorem	reldisj 4404	Two ways of saying that two classes are disjoint, using the complement of 𝐵 relative to a universe 𝐶. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid ax-12 2182. (Revised by GG, 28-Jun-2024.)
⊢ (𝐴 ⊆ 𝐶 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
Theorem	disj3 4405	Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))
Theorem	disjne 4406	Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷)
Theorem	disjeq0 4407	Two disjoint sets are equal iff both are empty. (Contributed by AV, 19-Jun-2022.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
Theorem	disjel 4408	A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶 ∈ 𝐵)
Theorem	disj2 4409	Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))
Theorem	disj4 4410	Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴)
Theorem	ssdisj 4411	Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)
Theorem	disjpss 4412	A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ 𝐵))
Theorem	undisj1 4413	The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
⊢ (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐶) = ∅) ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ∅)
Theorem	undisj2 4414	The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ∩ 𝐶) = ∅) ↔ (𝐴 ∩ (𝐵 ∪ 𝐶)) = ∅)
Theorem	ssindif0 4415	Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)
Theorem	inelcm 4416	The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅)
Theorem	minel 4417	A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶)
Theorem	undif4 4418	Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ∩ 𝐶) = ∅ → (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶))
Theorem	disjssun 4419	Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
Theorem	vdif0 4420	Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅)
Theorem	difrab0eq 4421*	If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
⊢ ((𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅ ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑})
Theorem	pssnel 4422*	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 4423	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 4424	A class and its relative complement are disjoint. (Contributed by Thierry Arnoux, 29-Nov-2023.)
⊢ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅
Theorem	difin0 4425	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 4426	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 4427	Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 4423). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
Theorem	undif2 4428	Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 4423). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)
⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
Theorem	undifabs 4429	Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴
Theorem	inundif 4430	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 4431	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 4432	Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
Theorem	undif 4433	Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
Theorem	undifr 4434	Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023.) (Proof shortened by SN, 11-Mar-2025.)
⊢ (𝐴 ⊆ 𝐵 ↔ ((𝐵 ∖ 𝐴) ∪ 𝐴) = 𝐵)
Theorem	undifrOLD 4435	Obsolete version of undifr 4434 as of 11-Mar-2025. (Contributed by Thierry Arnoux, 21-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 ↔ ((𝐵 ∖ 𝐴) ∪ 𝐴) = 𝐵)
Theorem	undif5 4436	An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴)
Theorem	ssdifin0 4437	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 4438	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 4439	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 4440	Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)
⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)
Theorem	pssdifcom1 4441	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 4442	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 4443	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 4444	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 4445*	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 4446*	Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1977). 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 4447*	A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4446. 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 4448*	Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	r19.28z 4449*	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 4450*	Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	r19.9rzv 4451*	Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	r19.28zv 4452*	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 4453*	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 4454*	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 4455*	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 4456*	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 4457*	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 4458*	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 4459*	Idempotent law for restricted quantifier. Weak version of ralidm 4463, which does not require ax-10 2146, ax-12 2182, but requires ax-8 2115. (Contributed by GG, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rzal 4460*	Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid df-clel 2808, ax-8 2115. (Revised by GG, 2-Sep-2024.)
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rzalALT 4461*	Alternate proof of rzal 4460. Shorter, but requiring df-clel 2808, ax-8 2115. (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 4462*	Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2808, ax-8 2115. (Revised by GG, 2-Sep-2024.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
Theorem	ralidm 4463	Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) Reduce axiom usage. (Revised by GG, 2-Sep-2024.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	ral0 4464	Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) Avoid df-clel 2808, ax-8 2115. (Revised by GG, 2-Sep-2024.)
⊢ ∀𝑥 ∈ ∅ 𝜑
Theorem	ralf0 4465*	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 2808, ax-8 2115. (Revised by GG, 2-Sep-2024.)
⊢ ¬ 𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)
Theorem	ralnralall 4466*	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 4467*	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 4468*	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 4469*	Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
Theorem	sbss 4470*	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 4471	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 4472*	Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4468. 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 4473*	Lemma for 2reu4 4474. (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 4474*	Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2652. (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 4475	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 4477). It is the analogue for classes of the conditional operator for propositions, denoted by if-(𝜑, 𝜓, 𝜒) (see df-ifp 1063).
Syntax	cif 4476	Extend class notation to include the conditional operator for classes.
class if(𝜑, 𝐴, 𝐵)
Definition	df-if 4477*	Definition of the conditional operator for classes. The expression if(𝜑, 𝐴, 𝐵) is read "if 𝜑 then 𝐴 else 𝐵". See iftrue 4482 and iffalse 4485 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 4535. (Contributed by NM, 15-May-1999.)
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
Theorem	dfif2 4478*	An alternate definition of the conditional operator df-if 4477 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 4479*	An alternate definition of the conditional operator df-if 4477 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
Theorem	ifeq1 4480	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 4481	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 4482	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 4483	Inference associated with iftrue 4482. (Contributed by BJ, 7-Oct-2018.)
⊢ 𝜑    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴
Theorem	iftrued 4484	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 4485	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 4486	Inference associated with iffalse 4485. (Contributed by BJ, 7-Oct-2018.)
⊢ ¬ 𝜑    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵
Theorem	iffalsed 4487	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 4488	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 4485 directly in this case. It happens, e.g., in oevn0 8439. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 ≠ 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
Theorem	iftrueb 4489	When the branches are not equal, an "if" condition results in the first branch if and only if its condition is true. (Contributed by SN, 16-Oct-2025.)
⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = 𝐴 ↔ 𝜑))
Theorem	ifsb 4490	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 4491*	Alternate definition of the conditional operator df-if 4477. 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 4492*	Alternate definition of the conditional operator df-if 4477. 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 4493*	Alternate definition of the conditional operator df-if 4477. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false (see also ab0orv 4334). (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 4494	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 4495	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 4496	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 4497	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 4498	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 4499	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Theorem	ifbid 4500	Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))