P. 39 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3801-3900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sbc2iedv 3801*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))
Theorem	sbc3ie 3802*	Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑 ↔ 𝜓)
Theorem	sbccomlem 3803*	Lemma for sbccom 3804. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Theorem	sbccom 3804*	Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Theorem	sbcralt 3805*	Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
Theorem	sbcrext 3806*	Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
Theorem	sbcralg 3807*	Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
Theorem	sbcrex 3808*	Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)
⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)
Theorem	sbcreu 3809*	Interchange class substitution and restricted unique existential quantifier. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 18-Aug-2018.)
⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)
Theorem	reu8nf 3810*	Restricted uniqueness using implicit substitution. This version of reu8 3668 uses a nonfreeness hypothesis for 𝑥 and 𝜓 instead of distinct variable conditions. (Contributed by AV, 21-Jan-2022.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
Theorem	sbcabel 3811*	Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]{𝑦 ∣ 𝜑} ∈ 𝐵 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∈ 𝐵))
Theorem	rspsbc 3812*	Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 2071 and spsbc 3729. See also rspsbca 3813 and rspcsbela 4369. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑))
Theorem	rspsbca 3813*	Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝜑) → [𝐴 / 𝑥]𝜑)
Theorem	rspesbca 3814*	Existence form of rspsbca 3813. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑)
Theorem	spesbc 3815	Existence form of spsbc 3729. (Contributed by Mario Carneiro, 18-Nov-2016.)
⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)
Theorem	spesbcd 3816	form of spsbc 3729. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → [𝐴 / 𝑥]𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	sbcth2 3817*	A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
⊢ (𝑥 ∈ 𝐵 → 𝜑)    ⇒   ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑)
Theorem	ra4v 3818*	Version of ra4 3819 with a disjoint variable condition, requiring fewer axioms. This is stdpc5v 1941 for a restricted domain. (Contributed by BJ, 27-Mar-2020.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ra4 3819	Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is the axiom stdpc5 2201 of standard predicate calculus for a restricted domain. See ra4v 3818 for a version requiring fewer axioms. (Contributed by NM, 16-Jan-2004.) (Proof shortened by BJ, 27-Mar-2020.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
Theorem	rmo2 3820*	Alternate definition of restricted "at most one". Note that ∃*𝑥 ∈ 𝐴𝜑 is not equivalent to ∃𝑦 ∈ 𝐴∀𝑥 ∈ 𝐴(𝜑 → 𝑥 = 𝑦) (in analogy to reu6 3661); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3821. (Contributed by NM, 17-Jun-2017.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
Theorem	rmo2i 3821*	Condition implying restricted "at most one". (Contributed by NM, 17-Jun-2017.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	rmo3 3822*	Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) Avoid ax-13 2372. (Revised by Wolf Lammen, 30-Apr-2023.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	rmob 3823*	Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
Theorem	rmoi 3824*	Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶)
Theorem	rmob2 3825*	Consequence of "restricted at most one". (Contributed by Thierry Arnoux, 9-Dec-2019.)
⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒))
Theorem	rmoi2 3826*	Consequence of "restricted at most one". (Contributed by Thierry Arnoux, 9-Dec-2019.)
⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → 𝑥 = 𝐵)
Theorem	rmoanim 3827	Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2622. (Contributed by Alexander van der Vekens, 25-Jun-2017.) Avoid ax-10 2137 and ax-11 2154. (Revised by Gino Giotto, 24-Aug-2023.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓))
Theorem	rmoanimALT 3828	Alternate proof of rmoanim 3827, shorter but requiring ax-10 2137 and ax-11 2154. (Contributed by Alexander van der Vekens, 25-Jun-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓))
Theorem	reuan 3829	Introduction of a conjunct into restricted unique existential quantifier, analogous to euan 2623. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓))
Theorem	2reu1 3830*	Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2652. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)))
Theorem	2reu2 3831*	Double restricted existential uniqueness, analogous to 2eu2 2654. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
2.1.10  Proper substitution of classes for sets into classes
Syntax	csb 3832	Extend class notation to include the proper substitution of a class for a set into another class.
class ⦋𝐴 / 𝑥⦌𝐵
Definition	df-csb 3833*	Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 3716, to prevent ambiguity. Theorem sbcel1g 4347 shows an example of how ambiguity could arise if we did not use distinguished brackets. When 𝐴 is a proper class, this evaluates to the empty set (see csbprc 4340). Theorem sbccsb 4367 recovers substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
Theorem	csb2 3834*	Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that 𝑥 can be free in 𝐵 but cannot occur in 𝐴. (Contributed by NM, 2-Dec-2013.)
⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)}
Theorem	csbeq1 3835	Analogue of dfsbcq 3718 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)
Theorem	csbeq1d 3836	Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)
Theorem	csbeq2 3837	Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
⊢ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
Theorem	csbeq2d 3838	Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
Theorem	csbeq2dv 3839*	Formula-building deduction for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
Theorem	csbeq2i 3840	Formula-building inference for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶
Theorem	csbeq12dv 3841*	Formula-building inference for class substitution. (Contributed by SN, 3-Nov-2023.)
⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)
Theorem	cbvcsbw 3842*	Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. Version of cbvcsb 3843 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷
Theorem	cbvcsb 3843	Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvcsbw 3842 when possible. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷
Theorem	cbvcsbv 3844*	Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶
Theorem	csbid 3845	Analogue of sbid 2248 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴
Theorem	csbeq1a 3846	Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
Theorem	csbcow 3847*	Composition law for chained substitutions into a class. Version of csbco 3848 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
Theorem	csbco 3848*	Composition law for chained substitutions into a class. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker csbcow 3847 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.)
⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
Theorem	csbtt 3849	Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
Theorem	csbconstgf 3850	Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by NM, 10-Nov-2005.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
Theorem	csbconstg 3851*	Substitution doesn't affect a constant 𝐵 (in which 𝑥 does not occur). csbconstgf 3850 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.) Avoid ax-12 2171. (Revised by Gino Giotto, 15-Oct-2024.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
Theorem	csbconstgOLD 3852*	Obsolete version of csbconstg 3851 as of 15-Oct-2024. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
Theorem	csbgfi 3853	Substitution for a variable not free in a class does not affect it, in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐵
Theorem	csbconstgi 3854*	The proper substitution of a class for a variable in another variable does not modify it, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝑦 = 𝑦
Theorem	nfcsb1d 3855	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
Theorem	nfcsb1 3856	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵
Theorem	nfcsb1v 3857*	Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵
Theorem	nfcsbd 3858	Deduction version of nfcsb 3860. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
Theorem	nfcsbw 3859*	Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3860 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵
Theorem	nfcsb 3860	Bound-variable hypothesis builder for substitution into a class. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfcsbw 3859 when possible. (Contributed by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵
Theorem	csbhypf 3861*	Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3491 for class substitution version. (Contributed by NM, 19-Dec-2008.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbiebt 3862*	Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3866.) (Contributed by NM, 11-Nov-2005.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
Theorem	csbiedf 3863*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbieb 3864*	Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴 → 𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbiebg 3865*	Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴 → 𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
Theorem	csbiegf 3866*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶)    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbief 3867*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
Theorem	csbie 3868*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
Theorem	csbieOLD 3869*	Obsolete version of csbie 3868 as of 15-Oct-2024. (Contributed by AV, 2-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
Theorem	csbied 3870*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbiedOLD 3871*	Obsolete version of csbied 3870 as of 15-Oct-2024. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbied2 3872*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷)
Theorem	csbie2t 3873*	Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3874). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷)
Theorem	csbie2 3874*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷
Theorem	csbie2g 3875*	Conversion of implicit substitution to explicit class substitution. This version of csbie 3868 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    &   ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)
Theorem	cbvrabcsfw 3876*	Version of cbvrabcsf 3880 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Andrew Salmon, 13-Jul-2011.) (Revised by Gino Giotto, 26-Jan-2024.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
Theorem	cbvralcsf 3877	A more general version of cbvralf 3371 that doesn't require 𝐴 and 𝐵 to be distinct from 𝑥 or 𝑦. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Andrew Salmon, 13-Jul-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)
Theorem	cbvrexcsf 3878	A more general version of cbvrexf 3372 that has no distinct variable restrictions. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)
Theorem	cbvreucsf 3879	A more general version of cbvreuv 3390 that has no distinct variable restrictions. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Andrew Salmon, 13-Jul-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓)
Theorem	cbvrabcsf 3880	A more general version of cbvrab 3425 with no distinct variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Andrew Salmon, 13-Jul-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
Theorem	cbvralv2 3881*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)
Theorem	cbvrexv2 3882*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)
Theorem	rspc2vd 3883*	Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class 𝐷 for the second set variable 𝑦 may depend on the first set variable 𝑥. (Contributed by AV, 29-Mar-2021.)
⊢ (𝑥 = 𝐴 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐸)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐸)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜃 → 𝜓))
2.1.11  Define basic set operations and relations
Syntax	cdif 3884	Extend class notation to include class difference (read: "𝐴 minus 𝐵").
class (𝐴 ∖ 𝐵)
Syntax	cun 3885	Extend class notation to include union of two classes (read: "𝐴 union 𝐵").
class (𝐴 ∪ 𝐵)
Syntax	cin 3886	Extend class notation to include the intersection of two classes (read: "𝐴 intersect 𝐵").
class (𝐴 ∩ 𝐵)
Syntax	wss 3887	Extend wff notation to include the subclass relation. This is read "𝐴 is a subclass of 𝐵 " or "𝐵 includes 𝐴". When 𝐴 exists as a set, it is also read "𝐴 is a subset of 𝐵".
wff 𝐴 ⊆ 𝐵
Syntax	wpss 3888	Extend wff notation with proper subclass relation.
wff 𝐴 ⊊ 𝐵
Theorem	difjust 3889*	Soundness justification theorem for df-dif 3890. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)}
Definition	df-dif 3890*	Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, ({1, 3} ∖ {1, 8}) = {3} (ex-dif 28787). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3892) and intersection (𝐴 ∩ 𝐵) (df-in 3894). Several notations are used in the literature; we chose the ∖ convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology "𝐴 excludes 𝐵 " to mean 𝐴 ∖ 𝐵. We will use "𝐵 is removed from 𝐴 " to mean 𝐴 ∖ {𝐵} i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
Theorem	unjust 3891*	Soundness justification theorem for df-un 3892. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
Definition	df-un 3892*	Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∪ {1, 8}) = {1, 3, 8} (ex-un 28788). Contrast this operation with difference (𝐴 ∖ 𝐵) (df-dif 3890) and intersection (𝐴 ∩ 𝐵) (df-in 3894). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 4193. For union defined in terms of intersection, see dfun3 4199. (Contributed by NM, 23-Aug-1993.)
⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
Theorem	injust 3893*	Soundness justification theorem for df-in 3894. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
Definition	df-in 3894*	Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 28789). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3892) and difference (𝐴 ∖ 𝐵) (df-dif 3890). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4194 and dfin4 4201. For intersection defined in terms of union, see dfin3 4200. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
Theorem	dfin5 3895*	Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}
Theorem	dfdif2 3896*	Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
Theorem	eldif 3897	Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
Theorem	eldifd 3898	If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3897. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))
Theorem	eldifad 3899	If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3897. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐵)
Theorem	eldifbd 3900	If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3897. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)