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Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbcimg 2801 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 
Theoremsbcan 2802 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
 
Theoremsbcang 2803 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 
Theoremsbcor 2804 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
 
Theoremsbcorg 2805 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 
Theoremsbcbig 2806 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 
Theoremsbcal 2807* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
 
Theoremsbcalg 2808* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
 
Theoremsbcex2 2809* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
 
Theoremsbcexg 2810* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
(𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
 
Theoremsbceqal 2811* A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
(𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
 
Theoremsbeqalb 2812* 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 2813 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 
Theoremsbcbidv 2814* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 
Theoremsbcbii 2815 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
(𝜑𝜓)       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)
 
Theoremeqsbc3r 2816* eqsbc3 2799 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
 
Theoremsbc3ang 2817 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 2818* 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 2819* 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 2820* 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 2821 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 2822 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 2823 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 2824* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2822. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))
 
Theoremsbc2iegf 2825* 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 2826* 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 2827* 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 2828* 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 2829* Lemma for sbccom 2830. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 
Theoremsbccom 2830* 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 2831* 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 2832* 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 2833* 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 2834* 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 2835* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
(𝐴𝑉 → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
 
Theoremsbcabel 2836* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
𝑥𝐵       (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))
 
Theoremrspsbc 2837* 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 2772. See also rspsbca 2838 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))
 
Theoremrspsbca 2838* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → [𝐴 / 𝑥]𝜑)
 
Theoremrspesbca 2839* Existence form of rspsbca 2838. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑥𝐵 𝜑)
 
Theoremspesbc 2840 Existence form of spsbc 2772. (Contributed by Mario Carneiro, 18-Nov-2016.)
([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)
 
Theoremspesbcd 2841 form of spsbc 2772. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑[𝐴 / 𝑥]𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theoremsbcth2 2842* 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 2843 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 2844* Condition implying restricted "at most one." (Contributed by Jim Kingdon, 14-Jul-2018.)
𝑦𝜑       (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
 
Theoremrmo2i 2845* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
𝑦𝜑       (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
 
Theoremrmo3 2846* 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 2847* 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 2848* 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 2849 Extend class notation to include the proper substitution of a class for a set into another class.
class 𝐴 / 𝑥𝐵
 
Definitiondf-csb 2850* 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 2761, to prevent ambiguity. Theorem sbcel1g 2866 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 2875 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
 
Theoremcsb2 2851* 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 2852 Analog of dfsbcq 2763 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(𝐴 = 𝐵𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐶)
 
Theoremcbvcsb 2853 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 2854* 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 2855 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
(𝜑𝐴 = 𝐵)       (𝜑𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐶)
 
Theoremcsbid 2856 Analog of sbid 1657 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
𝑥 / 𝑥𝐴 = 𝐴
 
Theoremcsbeq1a 2857 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(𝑥 = 𝐴𝐵 = 𝐴 / 𝑥𝐵)
 
Theoremcsbco 2858* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
 
Theoremcsbtt 2859 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 2860 Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by NM, 10-Nov-2005.)
𝑥𝐵       (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
 
Theoremcsbconstg 2861* Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). csbconstgf 2860 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
(𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
 
Theoremsbcel12g 2862 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 2863 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 2864 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
Theoremsbcne12g 2865 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
Theoremsbcel1g 2866* 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 2867* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
 
Theoremsbcel2g 2868* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
 
Theoremsbceq2g 2869* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐵 = 𝐴 / 𝑥𝐶))
 
Theoremcsbcomg 2870* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
((𝐴𝑉𝐵𝑊) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶)
 
Theoremcsbeq2d 2871 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 2872* 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 2873 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 2874 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 2875* 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 2876 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑}))
 
Theoremnfcsb1d 2877 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑𝑥𝐴 / 𝑥𝐵)
 
Theoremnfcsb1 2878 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴       𝑥𝐴 / 𝑥𝐵
 
Theoremnfcsb1v 2879* 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 2880 Deduction version of nfcsb 2881. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴 / 𝑦𝐵)
 
Theoremnfcsb 2881 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴 / 𝑦𝐵
 
Theoremcsbhypf 2882* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2600 for class substitution version. (Contributed by NM, 19-Dec-2008.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
 
Theoremcsbiebt 2883* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2887.) (Contributed by NM, 11-Nov-2005.)
((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
 
Theoremcsbiedf 2884* 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 2885* 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 2886* 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 2887* 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 2888* 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 2889* 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 2890* 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 2891* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2892). (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 2892* 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 2893* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2794 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 2894 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 2895 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 2896* 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 2897* 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)
 
Theoremcsbnest1g 2898 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)
 
Theoremcsbidmg 2899* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
(𝐴𝑉𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
 
Theoremsbcco3g 2900* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
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