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	sbcbi2 3801	Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) (Proof shortened by Wolf Lammen, 4-May-2023.) Avoid ax-10, ax-12. (Revised by Steven Nguyen, 5-May-2024.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))
Theorem	sbcbi2OLD 3802	Obsolete proof of sbcbi2 3801 as of 5-May-2024. (Contributed by Giovanni Mascellani, 9-Apr-2018.) (Proof shortened by Wolf Lammen, 4-May-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))
Theorem	sbcal 3803*	Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 18-Aug-2018.)
⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
Theorem	sbcex2 3804*	Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.)
⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
Theorem	sbceqal 3805*	Class version of one implication of equvelv 2034. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by SN, 26-Oct-2024.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
Theorem	sbceqalOLD 3806*	Obsolete version of sbceqal 3805 as of 26-Oct-2024. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
Theorem	sbeqalb 3807*	Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → 𝐴 = 𝐵))
Theorem	eqsbc2 3808*	Substitution for the right-hand side in an equality. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴))
Theorem	sbc3an 3809	Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓 ∧ [𝐴 / 𝑥]𝜒))
Theorem	sbcel1v 3810*	Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) Avoid ax-13 2370. (Revised by Wolf Lammen, 30-Apr-2023.)
⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)
Theorem	sbcel2gv 3811*	Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
Theorem	sbcel21v 3812*	Class substitution into a membership relation. One direction of sbcel2gv 3811 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵)
Theorem	sbcimdv 3813*	Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1812). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
Theorem	sbcimdvOLD 3814*	Obsolete version of sbcimdv 3813 as of 12-Oct-2024. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
Theorem	sbctt 3815	Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbcgf 3816	Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbc19.21g 3817	Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓)))
Theorem	sbcg 3818*	Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3816. (Contributed by Alan Sare, 10-Nov-2012.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbcgOLD 3819*	Obsolete version of sbcg 3818 as of 12-Oct-2024. (Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbcgfi 3820	Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)
Theorem	sbc2iegf 3821*	Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))
Theorem	sbc2ie 3822*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Gino Giotto, 12-Oct-2024.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)
Theorem	sbc2ieOLD 3823*	Obsolete version of sbc2ie 3822 as of 12-Oct-2024. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)
Theorem	sbc2iedv 3824*	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 3825*	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 3826*	Lemma for sbccom 3827. (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 3827*	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 3828*	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 3829*	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 3830*	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 3831*	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 3832*	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 3833*	Restricted uniqueness using implicit substitution. This version of reu8 3691 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 3834*	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 3835*	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 3752. See also rspsbca 3836 and rspcsbela 4395. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑))
Theorem	rspsbca 3836*	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 3837*	Existence form of rspsbca 3836. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑)
Theorem	spesbc 3838	Existence form of spsbc 3752. (Contributed by Mario Carneiro, 18-Nov-2016.)
⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)
Theorem	spesbcd 3839	form of spsbc 3752. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → [𝐴 / 𝑥]𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	sbcth2 3840*	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 3841*	Version of ra4 3842 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 3842	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 3841 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 3843*	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 3684); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3844. (Contributed by NM, 17-Jun-2017.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
Theorem	rmo2i 3844*	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 3845*	Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) Avoid ax-13 2370. (Revised by Wolf Lammen, 30-Apr-2023.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	rmob 3846*	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 3847*	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 3848*	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 3849*	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 3850	Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2620. (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 3851	Alternate proof of rmoanim 3850, 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 3852	Introduction of a conjunct into restricted unique existential quantifier, analogous to euan 2621. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓))
Theorem	2reu1 3853*	Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2650. (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 3854*	Double restricted existential uniqueness, analogous to 2eu2 2652. (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 3855	Extend class notation to include the proper substitution of a class for a set into another class.
class ⦋𝐴 / 𝑥⦌𝐵
Definition	df-csb 3856*	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 3739, to prevent ambiguity. Theorem sbcel1g 4373 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 4366). Theorem sbccsb 4393 recovers substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
Theorem	csb2 3857*	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 3858	Analogue of dfsbcq 3741 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)
Theorem	csbeq1d 3859	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 3860	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 3861	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 3862*	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 3863	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 3864*	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 3865*	Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. Version of cbvcsb 3866 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Jeff Hankins, 13-Sep-2009.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷
Theorem	cbvcsb 3866	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 2370. Use the weaker cbvcsbw 3865 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 3867*	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 3868	Analogue of sbid 2247 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴
Theorem	csbeq1a 3869	Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
Theorem	csbcow 3870*	Composition law for chained substitutions into a class. Version of csbco 3871 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 10-Nov-2005.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.)
⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
Theorem	csbco 3871*	Composition law for chained substitutions into a class. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker csbcow 3870 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.)
⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
Theorem	csbtt 3872	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 3873	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 3874*	Substitution doesn't affect a constant 𝐵 (in which 𝑥 does not occur). csbconstgf 3873 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 3875*	Obsolete version of csbconstg 3874 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 3876	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 3877*	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 3878	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 3879	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵
Theorem	nfcsb1v 3880*	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 3881	Deduction version of nfcsb 3883. Usage of this theorem is discouraged because it depends on ax-13 2370. (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 3882*	Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3883 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 12-Oct-2016.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵
Theorem	nfcsb 3883	Bound-variable hypothesis builder for substitution into a class. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfcsbw 3882 when possible. (Contributed by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵
Theorem	csbhypf 3884*	Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3507 for class substitution version. (Contributed by NM, 19-Dec-2008.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbiebt 3885*	Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3889.) (Contributed by NM, 11-Nov-2005.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
Theorem	csbiedf 3886*	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 3887*	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 3888*	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 3889*	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 3890*	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 3891*	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 3892*	Obsolete version of csbie 3891 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 3893*	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 3894*	Obsolete version of csbied 3893 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 3895*	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 3896*	Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3897). (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 3897*	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 3898*	Conversion of implicit substitution to explicit class substitution. This version of csbie 3891 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 3899*	Version of cbvrabcsf 3903 with a disjoint variable condition, which does not require ax-13 2370. (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 3900	A more general version of cbvralf 3333 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 2370. (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)