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	nfsbcd 3801	Deduction version of nfsbc 3802. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfsbcdw 3798 when possible. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Theorem	nfsbc 3802	Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfsbcw 3799 when possible. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑
Theorem	sbcco 3803*	A composition law for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker sbccow 3800 when possible. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) (New usage is discouraged.)
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
Theorem	sbcco2 3804*	A composition law for class substitution. Importantly, 𝑥 may occur free in the class expression substituted for 𝐴. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
Theorem	sbc5 3805*	An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by SN, 2-Sep-2024.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	sbc5ALT 3806*	Alternate proof of sbc5 3805. This proof helps show how clelab 2879 works, since it is equivalent but shorter thanks to now-available library theorems like vtoclbg 3559 and isset 3487. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	sbc6g 3807*	An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by SN, 5-Oct-2024.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
Theorem	sbc6gOLD 3808*	Obsolete version of sbc6g 3807 as of 5-Oct-2024. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
Theorem	sbc6 3809*	An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))
Theorem	sbc7 3810*	An equivalence for class substitution in the spirit of df-clab 2710. Note that 𝑥 and 𝐴 don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑))
Theorem	cbvsbcw 3811*	Change bound variables in a wff substitution. Version of cbvsbc 3813 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Jeff Hankins, 19-Sep-2009.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	cbvsbcvw 3812*	Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3814 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 30-Sep-2008.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	cbvsbc 3813	Change bound variables in a wff substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvsbcw 3811 when possible. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	cbvsbcv 3814*	Change the bound variable of a class substitution using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvsbcvw 3812 when possible. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	sbciegft 3815*	Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3816.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbciegf 3816*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbcieg 3817*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2137, ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbciegOLD 3818*	Obsolete version of sbcieg 3817 as of 12-Oct-2024. (Contributed by NM, 10-Nov-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbcie2g 3819*	Conversion of implicit substitution to explicit class substitution. This version of sbcie 3820 avoids a disjointness condition on 𝑥, 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒))
Theorem	sbcie 3820*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)
Theorem	sbciedf 3821*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbcied 3822*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) Avoid ax-10 2137, ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbciedOLD 3823*	Obsolete version of sbcied 3822 as of 12-Oct-2024. (Contributed by NM, 13-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbcied2 3824*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
Theorem	elrabsf 3825	Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3679 has implicit substitution). The hypothesis specifies that 𝑥 must not be a free variable in 𝐵. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
Theorem	eqsbc1 3826*	Substitution for the left-hand side in an equality. Class version of eqsb1 2859. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2371. (Revised by Wolf Lammen, 29-Apr-2023.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
Theorem	sbcng 3827	Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
Theorem	sbcimg 3828	Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
Theorem	sbcan 3829	Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))
Theorem	sbcor 3830	Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))
Theorem	sbcbig 3831	Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
Theorem	sbcn1 3832	Move negation in and out of class substitution. One direction of sbcng 3827 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)
Theorem	sbcim1 3833	Distribution of class substitution over implication. One direction of sbcimg 3828 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 26-Oct-2024.)
⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
Theorem	sbcim1OLD 3834	Obsolete version of sbcim1 3833 as of 26-Oct-2024. (Contributed by NM, 17-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
Theorem	sbcbid 3835	Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))
Theorem	sbcbidv 3836*	Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) Drop ax-12 2171. (Revised by Gino Giotto, 1-Dec-2023.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))
Theorem	sbcbii 3837	Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)
Theorem	sbcbi1 3838	Distribution of class substitution over biconditional. One direction of sbcbig 3831 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))
Theorem	sbcbi2 3839	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 3840	Obsolete proof of sbcbi2 3839 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 3841*	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 3842*	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 3843*	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 3844*	Obsolete version of sbceqal 3843 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 3845*	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 3846*	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 3847	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 3848*	Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) Avoid ax-13 2371. (Revised by Wolf Lammen, 30-Apr-2023.)
⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)
Theorem	sbcel2gv 3849*	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 3850*	Class substitution into a membership relation. One direction of sbcel2gv 3849 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵)
Theorem	sbcimdv 3851*	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 3852*	Obsolete version of sbcimdv 3851 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 3853	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 3854	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 3855	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 3856*	Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3854. (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 3857*	Obsolete version of sbcg 3856 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 3858	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 3859*	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 3860*	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 3861*	Obsolete version of sbc2ie 3860 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 3862*	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 3863*	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 3864*	Lemma for sbccom 3865. (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 3865*	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 3866*	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 3867*	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 3868*	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 3869*	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 3870*	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 3871*	Restricted uniqueness using implicit substitution. This version of reu8 3729 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 3872*	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 3873*	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 3790. See also rspsbca 3874 and rspcsbela 4435. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑))
Theorem	rspsbca 3874*	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 3875*	Existence form of rspsbca 3874. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑)
Theorem	spesbc 3876	Existence form of spsbc 3790. (Contributed by Mario Carneiro, 18-Nov-2016.)
⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)
Theorem	spesbcd 3877	form of spsbc 3790. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → [𝐴 / 𝑥]𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	sbcth2 3878*	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 3879*	Version of ra4 3880 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 3880	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 3879 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 3881*	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 3722); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3882. (Contributed by NM, 17-Jun-2017.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
Theorem	rmo2i 3882*	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 3883*	Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) Avoid ax-13 2371. (Revised by Wolf Lammen, 30-Apr-2023.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	rmob 3884*	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 3885*	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 3886*	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 3887*	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 3888	Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2616. (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 3889	Alternate proof of rmoanim 3888, 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 3890	Introduction of a conjunct into restricted unique existential quantifier, analogous to euan 2617. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓))
Theorem	2reu1 3891*	Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2646. (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 3892*	Double restricted existential uniqueness, analogous to 2eu2 2648. (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 3893	Extend class notation to include the proper substitution of a class for a set into another class.
class ⦋𝐴 / 𝑥⦌𝐵
Definition	df-csb 3894*	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 3777, to prevent ambiguity. Theorem sbcel1g 4413 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 4406). Theorem sbccsb 4433 recovers substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
Theorem	csb2 3895*	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 3896	Analogue of dfsbcq 3779 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)
Theorem	csbeq1d 3897	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 3898	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 3899	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 3900*	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)