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Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreuun1 3901* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)

Theoremreuun2 3902* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
(¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))

Theoremreupick 3903* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
(((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))

Theoremreupick3 3904* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))

Theoremreupick2 3905* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
(((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜑𝜓))

Theoremeuelss 3906* Transfer uniqueness of an element to a smaller subclass. (Contributed by AV, 14-Apr-2020.)
((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴 ∧ ∃!𝑥 𝑥𝐵) → ∃!𝑥 𝑥𝐴)

2.1.14  The empty set

Syntaxc0 3907 Extend class notation to include the empty set.
class

Definitiondf-nul 3908 Define the empty set. Special case of Exercise 4.10(o) of [Mendelson] p. 231. For a more traditional definition, but requiring a dummy variable, see dfnul2 3909. (Contributed by NM, 17-Jun-1993.)
∅ = (V ∖ V)

Theoremdfnul2 3909 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Theoremdfnul3 3910 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}

Theoremnoel 3911 The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
¬ 𝐴 ∈ ∅

Theoremn0i 3912 If a set has elements, then it is not empty. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴 → ¬ 𝐴 = ∅)

Theoremne0i 3913 If a set has elements, then it is not empty. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴𝐴 ≠ ∅)

Theoremn0ii 3914 If a set has elements, then it is not empty. Inference associated with n0i 3912. (Contributed by BJ, 15-Jul-2021.)
𝐴𝐵        ¬ 𝐵 = ∅

Theoremne0ii 3915 If a set has elements, then it is not empty. Inference associated with ne0i 3913. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴𝐵       𝐵 ≠ ∅

Theoremvn0 3916 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
V ≠ ∅

Theoremeq0f 3917 The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by BJ, 15-Jul-2021.)
𝑥𝐴       (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)

Theoremneq0f 3918 A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of neq0 3922 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by BJ, 15-Jul-2021.)
𝑥𝐴       𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)

Theoremn0f 3919 A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3923 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by NM, 17-Oct-2003.)
𝑥𝐴       (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)

Theoremn0fOLD 3920 Obsolete proof of n0f 3919 as of 15-Jul-2021. (Contributed by NM, 17-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝐴       (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)

Theoremeq0 3921* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)

Theoremneq0 3922* A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 21-Jun-1993.)
𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)

Theoremn0 3923* A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 29-Sep-2006.)
(𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)

Theoremnel0 3924* From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
¬ 𝑥𝐴       𝐴 = ∅

Theoremreximdva0 3925* Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)
((𝜑𝑥𝐴) → 𝜓)       ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)

Theoremrspn0 3926* Specialization for restricted generalization with a nonempty set. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
(𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))

Theoremn0rex 3927* There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.)
(𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)

Theoremssn0rex 3928* There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.)
((𝐴𝐵𝐴 ≠ ∅) → ∃𝑥𝐵 𝑥𝐴)

Theoremn0moeu 3929* A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
(𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))

Theoremrex0 3930 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
¬ ∃𝑥 ∈ ∅ 𝜑

Theorem0el 3931* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
(∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)

Theoremeqeuel 3932* A condition which implies the existence of a unique element of a class. (Contributed by AV, 4-Jan-2022.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥𝐴)

Theoremssdif0 3933 Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵 ↔ (𝐴𝐵) = ∅)

Theoremdifn0 3934 If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)
((𝐴𝐵) ≠ ∅ → 𝐴𝐵)

Theorempssdifn0 3935 A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)

Theorempssdif 3936 A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
(𝐴𝐵 → (𝐵𝐴) ≠ ∅)

Theoremdifin0ss 3937 Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
(((𝐴𝐵) ∩ 𝐶) = ∅ → (𝐶𝐴𝐶𝐵))

Theoreminssdif0 3938 Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
((𝐴𝐵) ⊆ 𝐶 ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)

Theoremdifid 3939 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.)
(𝐴𝐴) = ∅

TheoremdifidALT 3940 Alternate proof of difid 3939. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐴) = ∅

Theoremdif0 3941 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

Theoremab0 3942 The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 3945 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne 2792). (Contributed by BJ, 19-Mar-2021.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Theoremdfnf5 3943 Characterization of non-freeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.)
(Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = ∅ ∨ {𝑥𝜑} = V))

Theoremab0orv 3944* The class builder of a wff not containing the abstraction variable is either the empty set or the universal class. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.)
({𝑥𝜑} = ∅ ∨ {𝑥𝜑} = V)

Theoremabn0 3945 Nonempty class abstraction. See also ab0 3942. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)

Theoremrab0 3946 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.) (Proof shortened by JJ, 14-Jul-2021.)
{𝑥 ∈ ∅ ∣ 𝜑} = ∅

Theoremrab0OLD 3947 Obsolete proof of rab0 3946 as of 14-Jul-2021. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥 ∈ ∅ ∣ 𝜑} = ∅

Theoremrabeq0 3948 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.) (Revised by BJ, 16-Jul-2021.)
({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥𝐴 ¬ 𝜑)

Theoremrabn0 3949 Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.) (Revised by BJ, 16-Jul-2021.)
({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)

Theoremrabn0OLD 3950 Obsolete proof of rabn0 3949 as of 16-Jul-2021. (Contributed by NM, 29-Aug-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)

Theoremrabeq0OLD 3951 Obsolete proof of rabeq0 3948 as of 16-Jul-2021. (Contributed by Jeff Madsen, 7-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥𝐴 ¬ 𝜑)

Theoremrabxm 3952* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})

Theoremrabnc 3953* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅

Theoremelneldisj 3954* The set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element are disjoint. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.)
𝐸 = {𝑠𝐴𝐵𝐶}    &   𝑁 = {𝑠𝐴𝐵𝐶}       (𝐸𝑁) = ∅

Theoremelnelun 3955* The union of the set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element yields the original set. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.)
𝐸 = {𝑠𝐴𝐵𝐶}    &   𝑁 = {𝑠𝐴𝐵𝐶}       (𝐸𝑁) = 𝐴

TheoremelneldisjOLD 3956* Obsolete version of elneldisj 3954 as of 17-Dec-2021. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐸 = {𝑠𝐴𝐵𝑠}    &   𝑁 = {𝑠𝐴𝐵𝑠}       (𝐸𝑁) = ∅

TheoremelnelunOLD 3957* Obsolete version of elnelun 3955 as of 17-Dec-2021. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐸 = {𝑠𝐴𝐵𝑠}    &   𝑁 = {𝑠𝐴𝐵𝑠}       (𝐸𝑁) = 𝐴

Theoremun0 3958 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 15-Jul-1993.)
(𝐴 ∪ ∅) = 𝐴

Theoremin0 3959 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 3960 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∩ 𝐴) = ∅

Theoreminv1 3961 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 3962 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 3963 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 3964 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 3965 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
(𝐴 ⊆ ∅ → 𝐴 = ∅)

Theoremsseq0 3966 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 3967 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
((𝐴𝐵𝐴 ≠ ∅) → 𝐵 ≠ ∅)

Theorem0dif 3968 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 3969 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
¬ 𝜑       {𝑥𝜑} = ∅

Theoremeq0rdv 3970* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
(𝜑 → ¬ 𝑥𝐴)       (𝜑𝐴 = ∅)

Theoremcsbprc 3971 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 3972 Obsolete proof of csbprc 3971 as of 27-Aug-2021. (Contributed by NM, 17-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)

Theoremcsb0 3973 The proper substitution of a class into the empty set is empty. (Contributed by NM, 18-Aug-2018.)
𝐴 / 𝑥∅ = ∅

Theoremsbcel12 3974 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 3975 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 3976 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremsbcne12 3977 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 3978* 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 3979* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))

Theoremsbcel2 3980* 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 3981* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐵 = 𝐴 / 𝑥𝐶))

Theoremcsbeq2d 3982 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 3983* 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 3984 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 3985* 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 3986 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 3987 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 3988* 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 3989* 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 3990* 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 3991* 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 3992 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 3993* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) (Revised by NM, 18-Aug-2018.)
𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵

Theoremcsbvarg 3994 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 3995* 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 3996 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 3997* Special case related to rspsbc 3511. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)

Theoremsbnfc2 3998* Two ways of expressing "𝑥 is (effectively) not free in 𝐴." (Contributed by Mario Carneiro, 14-Oct-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)

Theoremcsbab 3999* 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 4000 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)

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