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Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremin0 4001 The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 21-Jun-1993.)
(𝐴 ∩ ∅) = ∅

Theorem0in 4002 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∩ 𝐴) = ∅

Theoreminv1 4003 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

Theoremunv 4004 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

Theorem0ss 4005 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.)
∅ ⊆ 𝐴

Theoremss0b 4006 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 = ∅)

Theoremss0 4007 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
(𝐴 ⊆ ∅ → 𝐴 = ∅)

Theoremsseq0 4008 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)

Theoremssn0 4009 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
((𝐴𝐵𝐴 ≠ ∅) → 𝐵 ≠ ∅)

Theorem0dif 4010 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) = ∅

Theoremabf 4011 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
¬ 𝜑       {𝑥𝜑} = ∅

Theoremeq0rdv 4012* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
(𝜑 → ¬ 𝑥𝐴)       (𝜑𝐴 = ∅)

Theoremcsbprc 4013 The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.) (Proof shortened by JJ, 27-Aug-2021.)
𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)

TheoremcsbprcOLD 4014 Obsolete proof of csbprc 4013 as of 27-Aug-2021. (Contributed by NM, 17-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)

Theoremcsb0 4015 The proper substitution of a class into the empty set is empty. (Contributed by NM, 18-Aug-2018.)
𝐴 / 𝑥∅ = ∅

Theoremsbcel12 4016 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Theoremsbceqg 4017 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

Theoremsbcnel12g 4018 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremsbcne12 4019 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Theoremsbcel1g 4020* Move proper substitution in and out of a membership relation. Note that the scope of [𝐴 / 𝑥] is the wff 𝐵𝐶, whereas the scope of 𝐴 / 𝑥 is the class 𝐵. (Contributed by NM, 10-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐶))

Theoremsbceq1g 4021* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))

Theoremsbcel2 4022* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶)

Theoremsbceq2g 4023* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐵 = 𝐴 / 𝑥𝐶))

Theoremcsbeq2d 4024 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
𝑥𝜑    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)

Theoremcsbeq2dv 4025* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
(𝜑𝐵 = 𝐶)       (𝜑𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)

Theoremcsbeq2i 4026 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
𝐵 = 𝐶       𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶

Theoremcsbcom 4027* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶

Theoremsbcnestgf 4028 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))

Theoremcsbnestgf 4029 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)

Theoremsbcnestg 4030* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))

Theoremcsbnestg 4031* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)

Theoremsbcco3g 4032* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))

Theoremcsbco3g 4033* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)

Theoremcsbnest1g 4034 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)

Theoremcsbidm 4035* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) (Revised by NM, 18-Aug-2018.)
𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵

Theoremcsbvarg 4036 The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥𝑥 = 𝐴)

Theoremsbccsb 4037* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝜑𝑦𝐴 / 𝑥{𝑦𝜑})

Theoremsbccsb2 4038 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑})

Theoremrspcsbela 4039* Special case related to rspsbc 3551. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)

Theoremsbnfc2 4040* Two ways of expressing "𝑥 is (effectively) not free in 𝐴." (Contributed by Mario Carneiro, 14-Oct-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)

Theoremcsbab 4041* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 19-Aug-2018.)
𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}

Theoremcsbun 4042 Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) (Revised by NM, 13-Sep-2018.)
𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Theoremcsbin 4043 Distribute proper substitution into a class through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) (Revised by NM, 18-Aug-2018.)
𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Theoremun00 4044 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)

Theoremvss 4045 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 ⊆ 𝐴𝐴 = V)

Theorem0pss 4046 The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.)
(∅ ⊊ 𝐴𝐴 ≠ ∅)

Theoremnpss0 4047 No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
¬ 𝐴 ⊊ ∅

Theoremnpss0OLD 4048 Obsolete proof of npss0 4047 as of 14-Jul-2021. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
¬ 𝐴 ⊊ ∅

Theorempssv 4049 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
(𝐴 ⊊ V ↔ ¬ 𝐴 = V)

Theoremdisj 4050* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)

Theoremdisjr 4051* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)

Theoremdisj1 4052* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))

Theoremreldisj 4053 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.)
(𝐴𝐶 → ((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶𝐵)))

Theoremdisj3 4054 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))

Theoremdisjne 4055 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)

Theoremdisjel 4056 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)

Theoremdisj2 4057 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))

Theoremdisj4 4058 Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)
((𝐴𝐵) = ∅ ↔ ¬ (𝐴𝐵) ⊊ 𝐴)

Theoremssdisj 4059 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.) (Proof shortened by JJ, 14-Jul-2021.)
((𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)

TheoremssdisjOLD 4060 Obsolete proof of ssdisj 4059 as of 14-Jul-2021. (Contributed by FL, 24-Jan-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)

Theoremdisjpss 4061 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
(((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴𝐵))

Theoremundisj1 4062 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
(((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐵) ∩ 𝐶) = ∅)

Theoremundisj2 4063 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
(((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)

Theoremssindif0 4064 Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
(𝐴𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Theoreminelcm 4065 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
((𝐴𝐵𝐴𝐶) → (𝐵𝐶) ≠ ∅)

Theoremminel 4066 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.)
((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)

TheoremminelOLD 4067 Obsolete proof of minel 4066 as of 14-Jul-2021. (Contributed by NM, 22-Jun-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)

Theoremundif4 4068 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))

Theoremdisjssun 4069 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐶))

Theoremvdif0 4070 Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.)
(𝐴 = V ↔ (V ∖ 𝐴) = ∅)

Theoremdifrab0eq 4071* 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 ∖ {𝑥𝑉𝜑}) = ∅ ↔ 𝑉 = {𝑥𝑉𝜑})

Theorempssnel 4072* A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.)
(𝐴𝐵 → ∃𝑥(𝑥𝐵 ∧ ¬ 𝑥𝐴))

Theoremdisjdif 4073 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
(𝐴 ∩ (𝐵𝐴)) = ∅

Theoremdifin0 4074 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.)
((𝐴𝐵) ∖ 𝐵) = ∅

Theoremunvdif 4075 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

Theoremundif1 4076 Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 4073). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)

Theoremundif2 4077 Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 4073). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)
(𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)

Theoremundifabs 4078 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴

Theoreminundif 4079 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.)
((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴

Theoremdisjdif2 4080 The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)
((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)

Theoremdifun2 4081 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Theoremundif 4082 Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) = 𝐵)

Theoremssdifin0 4083 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 ⊆ (𝐵𝐶) → (𝐴𝐶) = ∅)

Theoremssdifeq0 4084 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
(𝐴 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)

Theoremssundif 4085 A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)
(𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)

Theoremdifcom 4086 Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)
((𝐴𝐵) ⊆ 𝐶 ↔ (𝐴𝐶) ⊆ 𝐵)

Theorempssdifcom1 4087 Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
((𝐴𝐶𝐵𝐶) → ((𝐶𝐴) ⊊ 𝐵 ↔ (𝐶𝐵) ⊊ 𝐴))

Theorempssdifcom2 4088 Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
((𝐴𝐶𝐵𝐶) → (𝐵 ⊊ (𝐶𝐴) ↔ 𝐴 ⊊ (𝐶𝐵)))

Theoremdifdifdir 4089 Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Theoremuneqdifeq 4090 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.)
((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))

TheoremuneqdifeqOLD 4091 Obsolete proof of uneqdifeq 4090 as of 14-Jul-2021. (Contributed by FL, 17-Nov-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))

Theoremraldifeq 4092* Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.)
(𝜑𝐴𝐵)    &   (𝜑 → ∀𝑥 ∈ (𝐵𝐴)𝜓)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))

Theoremr19.2z 4093* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1949). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)

Theoremr19.2zb 4094* A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4093. Interestingly enough, 𝜑 does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)
(𝐴 ≠ ∅ ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))

Theoremr19.3rz 4095* Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
𝑥𝜑       (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))

Theoremr19.28z 4096* 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.)
𝑥𝜑       (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))

Theoremr19.3rzv 4097* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
(𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))

Theoremr19.9rzv 4098* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
(𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥𝐴 𝜑))

Theoremr19.28zv 4099* 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 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))

Theoremr19.37zv 4100* 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 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃𝑥𝐴 𝜓)))

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