P. 36 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3501-3600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ne0ii 3501	If a class has elements, then it is nonempty. Inference associated with ne0i 3498. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐵 ≠ ∅
Theorem	vn0 3502	The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
⊢ V ≠ ∅
Theorem	vn0m 3503	The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)
⊢ ∃𝑥 𝑥 ∈ V
Theorem	n0rf 3504	An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class 𝐴 nonempty if 𝐴 ≠ ∅ and inhabited if it has at least one element. In classical logic these two concepts are equivalent, for example see Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0r 3505 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by Jim Kingdon, 31-Jul-2018.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
Theorem	n0r 3505*	An inhabited class is nonempty. See n0rf 3504 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
Theorem	neq0r 3506*	An inhabited class is nonempty. See n0rf 3504 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ 𝐴 = ∅)
Theorem	reximdva0m 3507*	Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ ((𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	n0mmoeu 3508*	A case of equivalence of "at most one" and "only one". If a class is inhabited, that class having at most one element is equivalent to it having only one element. (Contributed by Jim Kingdon, 31-Jul-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴))
Theorem	rex0 3509	Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
Theorem	eq0 3510*	The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
Theorem	eqv 3511*	The universe contains every set. (Contributed by NM, 11-Sep-2006.)
⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	notm0 3512*	A class is not inhabited if and only if it is empty. (Contributed by Jim Kingdon, 1-Jul-2022.)
⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅)
Theorem	nel0 3513*	From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
⊢ ¬ 𝑥 ∈ 𝐴    ⇒   ⊢ 𝐴 = ∅
Theorem	0el 3514*	Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	abvor0dc 3515*	The class builder of a decidable proposition not containing the abstraction variable is either the universal class or the empty set. (Contributed by Jim Kingdon, 1-Aug-2018.)
⊢ (DECID 𝜑 → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))
Theorem	abn0r 3516	Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
⊢ (∃𝑥𝜑 → {𝑥 ∣ 𝜑} ≠ ∅)
Theorem	abn0m 3517*	Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)
⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
Theorem	rabn0r 3518	Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅)
Theorem	rabn0m 3519*	Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
⊢ (∃𝑦 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
Theorem	rab0 3520	Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Theorem	rabeq0 3521	Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
Theorem	abeq0 3522	Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
Theorem	rabxmdc 3523*	Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
⊢ (∀𝑥DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
Theorem	rabnc 3524*	Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅
Theorem	un0 3525	The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ∪ ∅) = 𝐴
Theorem	in0 3526	The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ∩ ∅) = ∅
Theorem	0in 3527	The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (∅ ∩ 𝐴) = ∅
Theorem	inv1 3528	The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
⊢ (𝐴 ∩ V) = 𝐴
Theorem	unv 3529	The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
⊢ (𝐴 ∪ V) = V
Theorem	0ss 3530	The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
⊢ ∅ ⊆ 𝐴
Theorem	ss0b 3531	Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
Theorem	ss0 3532	Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
Theorem	sseq0 3533	A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅)
Theorem	ssn0 3534	A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐵 ≠ ∅)
Theorem	abf 3535	A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
⊢ ¬ 𝜑    ⇒   ⊢ {𝑥 ∣ 𝜑} = ∅
Theorem	eq0rdv 3536*	Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = ∅)
Theorem	csbprc 3537	The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)
⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
Theorem	un00 3538	Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)
Theorem	vss 3539	Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
Theorem	disj 3540*	Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
Theorem	disjr 3541*	Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ 𝐴)
Theorem	disj1 3542*	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 3543	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.)
⊢ (𝐴 ⊆ 𝐶 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
Theorem	disj3 3544	Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))
Theorem	disjne 3545	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	disjel 3546	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 3547	Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))
Theorem	ssdisj 3548	Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)
Theorem	undisj1 3549	The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
⊢ (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐶) = ∅) ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ∅)
Theorem	undisj2 3550	The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ∩ 𝐶) = ∅) ↔ (𝐴 ∩ (𝐵 ∪ 𝐶)) = ∅)
Theorem	ssindif0im 3551	Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)
Theorem	inelcm 3552	The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅)
Theorem	minel 3553	A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶)
Theorem	undif4 3554	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 3555	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	ssdif0im 3556	Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐵) = ∅)
Theorem	vdif0im 3557	Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
⊢ (𝐴 = V → (V ∖ 𝐴) = ∅)
Theorem	difrab0eqim 3558*	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 Jim Kingdon, 3-Aug-2018.)
⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑} → (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅)
Theorem	inssdif0im 3559	Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 → (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅)
Theorem	difid 3560	The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)
⊢ (𝐴 ∖ 𝐴) = ∅
Theorem	difidALT 3561	The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3560. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∖ 𝐴) = ∅
Theorem	dif0 3562	The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∖ ∅) = 𝐴
Theorem	0dif 3563	The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
⊢ (∅ ∖ 𝐴) = ∅
Theorem	disjdif 3564	A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
Theorem	difin0 3565	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	undif1ss 3566	Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐴 ∪ 𝐵)
Theorem	undif2ss 3567	Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ (𝐴 ∪ 𝐵)
Theorem	undifabs 3568	Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴
Theorem	inundifss 3569	The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴
Theorem	disjdif2 3570	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 3571	Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
Theorem	undifss 3572	Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵)
Theorem	ssdifin0 3573	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 3574	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	ssundifim 3575	A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ∖ 𝐵) ⊆ 𝐶)
Theorem	difdifdirss 3576	Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) ⊆ ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))
Theorem	uneqdifeqim 3577	Two ways that 𝐴 and 𝐵 can "partition" 𝐶 (when 𝐴 and 𝐵 don't overlap and 𝐴 is a part of 𝐶). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 → (𝐶 ∖ 𝐴) = 𝐵))
Theorem	r19.2m 3578*	Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1684). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)
⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
Theorem	r19.2mOLD 3579*	Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1684). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) Obsolete version of r19.2m 3578 as of 7-Apr-2023. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
Theorem	r19.3rm 3580*	Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	r19.28m 3581*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.3rmv 3582*	Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	r19.9rmv 3583*	Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	r19.28mv 3584*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.45mv 3585*	Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.44mv 3586*	Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)))
Theorem	r19.27m 3587*	Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))
Theorem	r19.27mv 3588*	Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))
Theorem	rzal 3589*	Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexn0 3590*	Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3591). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
Theorem	rexm 3591*	Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
Theorem	ralidm 3592*	Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	ral0 3593	Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
⊢ ∀𝑥 ∈ ∅ 𝜑
Theorem	ralf0 3594*	The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
⊢ ¬ 𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)
Theorem	ralm 3595	Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	raaanlem 3596*	Special case of raaan 3597 where 𝐴 is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
Theorem	raaan 3597*	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 3598*	Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
Theorem	sbss 3599*	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 3600	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))