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Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelrabsf 2801 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2696 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.)
𝑥𝐵       (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵[𝐴 / 𝑥]𝜑))

Theoremeqsbc3 2802* Substitution applied to an atomic wff. Set theory version of eqsb3 2141. (Contributed by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))

Theoremsbcng 2803 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))

Theoremsbcimg 2804 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Theoremsbcan 2805 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbcang 2806 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Theoremsbcor 2807 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbcorg 2808 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Theoremsbcbig 2809 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Theoremsbcal 2810* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)

Theoremsbcalg 2811* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))

Theoremsbcex2 2812* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)

Theoremsbcexg 2813* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
(𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))

Theoremsbceqal 2814* A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
(𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))

Theoremsbeqalb 2815* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
(𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝜑𝑥 = 𝐵)) → 𝐴 = 𝐵))

Theoremsbcbid 2816 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Theoremsbcbidv 2817* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Theoremsbcbii 2818 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
(𝜑𝜓)       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)

Theoremeqsbc3r 2819* eqsbc3 2802 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))

Theoremsbc3ang 2820 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))

Theoremsbcel1gv 2821* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))

Theoremsbcel2gv 2822* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐵𝑉 → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))

Theoremsbcimdv 2823* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
(𝜑 → (𝜓𝜒))       ((𝜑𝐴𝑉) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Theoremsbctt 2824 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))

Theoremsbcgf 2825 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.)
𝑥𝜑       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))

Theoremsbc19.21g 2826 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
𝑥𝜑       (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))

Theoremsbcg 2827* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2825. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))

Theoremsbc2iegf 2828* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
𝑥𝜓    &   𝑦𝜓    &   𝑥 𝐵𝑊    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))

Theoremsbc2ie 2829* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)

Theoremsbc2iedv 2830* 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 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))       (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))

Theoremsbc3ie 2831* 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 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑𝜓)

Theoremsbccomlem 2832* Lemma for sbccom 2833. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Theoremsbccom 2833* 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)

Theoremsbcralt 2834* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))

Theoremsbcrext 2835* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))

Theoremsbcralg 2836* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))

Theoremsbcrex 2837* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)

Theoremsbcreug 2838* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
(𝐴𝑉 → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))

Theoremsbcabel 2839* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
𝑥𝐵       (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))

Theoremrspsbc 2840* 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 1658 and spsbc 2775. See also rspsbca 2841 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))

Theoremrspsbca 2841* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → [𝐴 / 𝑥]𝜑)

Theoremrspesbca 2842* Existence form of rspsbca 2841. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑥𝐵 𝜑)

Theoremspesbc 2843 Existence form of spsbc 2775. (Contributed by Mario Carneiro, 18-Nov-2016.)
([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Theoremspesbcd 2844 form of spsbc 2775. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑[𝐴 / 𝑥]𝜓)       (𝜑 → ∃𝑥𝜓)

Theoremsbcth2 2845* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(𝑥𝐵𝜑)       (𝐴𝐵[𝐴 / 𝑥]𝜑)

Theoremra5 2846 Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1476. (Contributed by NM, 16-Jan-2004.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥𝐴 𝜓))

Theoremrmo2ilem 2847* Condition implying restricted "at most one." (Contributed by Jim Kingdon, 14-Jul-2018.)
𝑦𝜑       (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)

Theoremrmo2i 2848* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
𝑦𝜑       (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)

Theoremrmo3 2849* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
𝑦𝜑       (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Theoremrmob 2850* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
(𝑥 = 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝐶 → (𝜑𝜒))       ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))

Theoremrmoi 2851* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
(𝑥 = 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝐶 → (𝜑𝜒))       ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓) ∧ (𝐶𝐴𝜒)) → 𝐵 = 𝐶)

2.1.10  Proper substitution of classes for sets into classes

Syntaxcsb 2852 Extend class notation to include the proper substitution of a class for a set into another class.
class 𝐴 / 𝑥𝐵

Definitiondf-csb 2853* 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 2764, to prevent ambiguity. Theorem sbcel1g 2869 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 2878 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}

Theoremcsb2 2854* 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 / 𝑥𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴𝑦𝐵)}

Theoremcsbeq1 2855 Analog of dfsbcq 2766 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(𝐴 = 𝐵𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐶)

Theoremcbvcsb 2856 Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑦𝐶    &   𝑥𝐷    &   (𝑥 = 𝑦𝐶 = 𝐷)       𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷

Theoremcbvcsbv 2857* 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 = 𝑦𝐵 = 𝐶)       𝐴 / 𝑥𝐵 = 𝐴 / 𝑦𝐶

Theoremcsbeq1d 2858 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
(𝜑𝐴 = 𝐵)       (𝜑𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐶)

Theoremcsbid 2859 Analog of sbid 1657 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
𝑥 / 𝑥𝐴 = 𝐴

Theoremcsbeq1a 2860 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(𝑥 = 𝐴𝐵 = 𝐴 / 𝑥𝐵)

Theoremcsbco 2861* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵

Theoremcsbtt 2862 Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
((𝐴𝑉𝑥𝐵) → 𝐴 / 𝑥𝐵 = 𝐵)

Theoremcsbconstgf 2863 Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by NM, 10-Nov-2005.)
𝑥𝐵       (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)

Theoremcsbconstg 2864* Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). csbconstgf 2863 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
(𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)

Theoremsbcel12g 2865 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremsbceqg 2866 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 2867 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremsbcne12g 2868 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremsbcel1g 2869* 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 2870* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))

Theoremsbcel2g 2871* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))

Theoremsbceq2g 2872* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐵 = 𝐴 / 𝑥𝐶))

Theoremcsbcomg 2873* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
((𝐴𝑉𝐵𝑊) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶)

Theoremcsbeq2d 2874 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 2875* 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 2876 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

Theoremcsbvarg 2877 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)

Theoremsbccsbg 2878* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝑦𝐴 / 𝑥{𝑦𝜑}))

Theoremsbccsb2g 2879 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑}))

Theoremnfcsb1d 2880 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑𝑥𝐴 / 𝑥𝐵)

Theoremnfcsb1 2881 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴       𝑥𝐴 / 𝑥𝐵

Theoremnfcsb1v 2882* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴 / 𝑥𝐵

Theoremnfcsbd 2883 Deduction version of nfcsb 2884. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴 / 𝑦𝐵)

Theoremnfcsb 2884 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴 / 𝑦𝐵

Theoremcsbhypf 2885* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2603 for class substitution version. (Contributed by NM, 19-Dec-2008.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)

Theoremcsbiebt 2886* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2890.) (Contributed by NM, 11-Nov-2005.)
((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))

Theoremcsbiedf 2887* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
𝑥𝜑    &   (𝜑𝑥𝐶)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)       (𝜑𝐴 / 𝑥𝐵 = 𝐶)

Theoremcsbieb 2888* Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
𝐴 ∈ V    &   𝑥𝐶       (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶)

Theoremcsbiebg 2889* Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝐶       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))

Theoremcsbiegf 2890* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝐴𝑉𝑥𝐶)    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)

Theoremcsbief 2891* 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    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       𝐴 / 𝑥𝐵 = 𝐶

Theoremcsbied 2892* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)       (𝜑𝐴 / 𝑥𝐵 = 𝐶)

Theoremcsbied2 2893* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐴𝑉)    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑥 = 𝐵) → 𝐶 = 𝐷)       (𝜑𝐴 / 𝑥𝐶 = 𝐷)

Theoremcsbie2t 2894* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2895). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷)

Theoremcsbie2 2895* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)       𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷

Theoremcsbie2g 2896* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2797 avoids a disjointness condition on 𝑥 and 𝐴 by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   (𝑦 = 𝐴𝐶 = 𝐷)       (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐷)

Theoremsbcnestgf 2897 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 2898 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 2899* 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 2900* 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)

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