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Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfsbc1 2801 Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴       𝑥[𝐴 / 𝑥]𝜑
 
Theoremnfsbc1v 2802* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥[𝐴 / 𝑥]𝜑
 
Theoremnfsbcd 2803 Deduction version of nfsbc 2804. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
 
Theoremnfsbc 2804 Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴    &   𝑥𝜑       𝑥[𝐴 / 𝑦]𝜑
 
Theoremsbcco 2805* A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
 
Theoremsbcco2 2806* 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 = 𝑦𝐴 = 𝐵)       ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
 
Theoremsbc5 2807* An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
 
Theoremsbc6g 2808* An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
 
Theoremsbc6 2809* 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 = 𝐴𝜑))
 
Theoremsbc7 2810* An equivalence for class substitution in the spirit of df-clab 2041. 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))
 
Theoremcbvsbc 2811 Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
 
Theoremcbvsbcv 2812* Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝑥 = 𝑦 → (𝜑𝜓))       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
 
Theoremsbciegft 2813* Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2814.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
 
Theoremsbciegf 2814* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
 
Theoremsbcieg 2815* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
 
Theoremsbcie2g 2816* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2817 avoids a disjointness condition on 𝑥 and 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑦 = 𝐴 → (𝜓𝜒))       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜒))
 
Theoremsbcie 2817* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       ([𝐴 / 𝑥]𝜑𝜓)
 
Theoremsbciedf 2818* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
 
Theoremsbcied 2819* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
 
Theoremsbcied2 2820* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
 
Theoremelrabsf 2821 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2716 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 2822* Substitution applied to an atomic wff. Set theory version of eqsb3 2155. (Contributed by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
 
Theoremsbcng 2823 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
 
Theoremsbcimg 2824 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 
Theoremsbcan 2825 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
 
Theoremsbcang 2826 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 
Theoremsbcor 2827 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
 
Theoremsbcorg 2828 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 
Theoremsbcbig 2829 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 
Theoremsbcn1 2830 Move negation in and out of class substitution. One direction of sbcng 2823 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)
 
Theoremsbcim1 2831 Distribution of class substitution over implication. One direction of sbcimg 2824 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
 
Theoremsbcbi1 2832 Distribution of class substitution over biconditional. One direction of sbcbig 2829 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
 
Theoremsbcbi2 2833 Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
(∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
 
Theoremsbcal 2834* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
 
Theoremsbcalg 2835* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
 
Theoremsbcex2 2836* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
 
Theoremsbcexg 2837* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
(𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
 
Theoremsbceqal 2838* A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
(𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
 
Theoremsbeqalb 2839* 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 2840 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 
Theoremsbcbidv 2841* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 
Theoremsbcbii 2842 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
(𝜑𝜓)       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)
 
Theoremeqsbc3r 2843* eqsbc3 2822 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))
 
Theoremsbc3an 2844 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)
([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 
Theoremsbcel1v 2845* Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
 
Theoremsbcel2gv 2846* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐵𝑉 → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
 
Theoremsbcel21v 2847* Class substitution into a membership relation. One direction of sbcel2gv 2846 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐵 / 𝑥]𝐴𝑥𝐴𝐵)
 
Theoremsbcimdv 2848* Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1360). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 
Theoremsbctt 2849 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 2850 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 2851 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 2852* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2850. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))
 
Theoremsbc2iegf 2853* 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 2854* 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 2855* 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 2856* 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 2857* Lemma for sbccom 2858. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 
Theoremsbccom 2858* 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 2859* 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 2860* 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 2861* 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 2862* 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 2863* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
(𝐴𝑉 → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
 
Theoremsbcabel 2864* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
𝑥𝐵       (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))
 
Theoremrspsbc 2865* 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 1672 and spsbc 2795. See also rspsbca 2866 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))
 
Theoremrspsbca 2866* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → [𝐴 / 𝑥]𝜑)
 
Theoremrspesbca 2867* Existence form of rspsbca 2866. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑥𝐵 𝜑)
 
Theoremspesbc 2868 Existence form of spsbc 2795. (Contributed by Mario Carneiro, 18-Nov-2016.)
([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)
 
Theoremspesbcd 2869 form of spsbc 2795. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑[𝐴 / 𝑥]𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theoremsbcth2 2870* 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 2871 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 1490. (Contributed by NM, 16-Jan-2004.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥𝐴 𝜓))
 
Theoremrmo2ilem 2872* Condition implying restricted "at most one." (Contributed by Jim Kingdon, 14-Jul-2018.)
𝑦𝜑       (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
 
Theoremrmo2i 2873* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
𝑦𝜑       (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
 
Theoremrmo3 2874* 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 2875* 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 2876* 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 2877 Extend class notation to include the proper substitution of a class for a set into another class.
class 𝐴 / 𝑥𝐵
 
Definitiondf-csb 2878* 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 2784, to prevent ambiguity. Theorem sbcel1g 2894 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 2903 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
 
Theoremcsb2 2879* 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 2880 Analog of dfsbcq 2786 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(𝐴 = 𝐵𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐶)
 
Theoremcbvcsb 2881 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 2882* 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 2883 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
(𝜑𝐴 = 𝐵)       (𝜑𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐶)
 
Theoremcsbid 2884 Analog of sbid 1671 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
𝑥 / 𝑥𝐴 = 𝐴
 
Theoremcsbeq1a 2885 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(𝑥 = 𝐴𝐵 = 𝐴 / 𝑥𝐵)
 
Theoremcsbco 2886* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
 
Theoremcsbtt 2887 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 2888 Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by NM, 10-Nov-2005.)
𝑥𝐵       (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
 
Theoremcsbconstg 2889* Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). csbconstgf 2888 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
(𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
 
Theoremsbcel12g 2890 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 2891 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 2892 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
Theoremsbcne12g 2893 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
Theoremsbcel1g 2894* 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 2895* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
 
Theoremsbcel2g 2896* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
 
Theoremsbceq2g 2897* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐵 = 𝐴 / 𝑥𝐶))
 
Theoremcsbcomg 2898* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
((𝐴𝑉𝐵𝑊) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶)
 
Theoremcsbeq2d 2899 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 2900* 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)
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