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Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelabg 2801* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab2g 2802* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝑉 → (𝐴𝐵𝜓))
 
Theoremelab2 2803* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝐵𝜓)
 
Theoremelab4g 2804* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
 
Theoremelab3gf 2805 Membership in a class abstraction, with a weaker antecedent than elabgf 2798. (Contributed by NM, 6-Sep-2011.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab3g 2806* Membership in a class abstraction, with a weaker antecedent than elabg 2801. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab3 2807* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
(𝜓𝐴 ∈ V)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 
Theoremelrabi 2808* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
(𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)
 
Theoremelrabf 2809 Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
 
Theoremelrab3t 2810* Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2812.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
 
Theoremelrab 2811* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
 
Theoremelrab3 2812* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
 
Theoremelrabd 2813* Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2811. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝑥 = 𝐴 → (𝜓𝜒))    &   (𝜑𝐴𝐵)    &   (𝜑𝜒)       (𝜑𝐴 ∈ {𝑥𝐵𝜓})
 
Theoremelrab2 2814* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐶 = {𝑥𝐵𝜑}       (𝐴𝐶 ↔ (𝐴𝐵𝜓))
 
Theoremralab 2815* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝜓𝜒))
 
Theoremralrab 2816* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∀𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∀𝑥𝐴 (𝜓𝜒))
 
Theoremrexab 2817* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝜓𝜒))
 
Theoremrexrab 2818* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∃𝑥𝐴 (𝜓𝜒))
 
Theoremralab2 2819* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑦(𝜑𝜒))
 
Theoremralrab2 2820* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑦𝐴 (𝜑𝜒))
 
Theoremrexab2 2821* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
 
Theoremrexrab2 2822* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑦𝐴 (𝜑𝜒))
 
Theoremabidnf 2823* Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)
(𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
 
Theoremdedhb 2824* A deduction theorem for converting the inference 𝑥𝐴 => 𝜑 into a closed theorem. Use nfa1 1504 and nfab 2261 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 2823 is useful. (Contributed by NM, 8-Dec-2006.)
(𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴} → (𝜑𝜓))    &   𝜓       (𝑥𝐴𝜑)
 
Theoremeqeu 2825* A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑)
 
Theoremeueq 2826* Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
(𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
 
Theoremeueq1 2827* Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
𝐴 ∈ V       ∃!𝑥 𝑥 = 𝐴
 
Theoremeueq2dc 2828* Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (DECID 𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
 
Theoremeueq3dc 2829* Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &    ¬ (𝜑𝜓)       (DECID 𝜑 → (DECID 𝜓 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
 
Theoremmoeq 2830* There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
∃*𝑥 𝑥 = 𝐴
 
Theoremmoeq3dc 2831* "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &    ¬ (𝜑𝜓)       (DECID 𝜑 → (DECID 𝜓 → ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
 
Theoremmosubt 2832* "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)
(∀𝑦∃*𝑥𝜑 → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))
 
Theoremmosub 2833* "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
∃*𝑥𝜑       ∃*𝑥𝑦(𝑦 = 𝐴𝜑)
 
Theoremmo2icl 2834* Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
(∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
 
Theoremmob2 2835* Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
 
Theoremmoi2 2836* Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)
(𝑥 = 𝐴 → (𝜑𝜓))       (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑𝜓)) → 𝑥 = 𝐴)
 
Theoremmob 2837* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
 
Theoremmoi 2838* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓𝜒)) → 𝐴 = 𝐵)
 
Theoremmorex 2839* Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐵 ∈ V    &   (𝑥 = 𝐵 → (𝜑𝜓))       ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓𝐵𝐴))
 
Theoremeuxfr2dc 2840* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.)
𝐴 ∈ V    &   ∃*𝑦 𝑥 = 𝐴       (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑))
 
Theoremeuxfrdc 2841* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.)
𝐴 ∈ V    &   ∃!𝑦 𝑥 = 𝐴    &   (𝑥 = 𝐴 → (𝜑𝜓))       (DECID𝑦𝑥(𝑥 = 𝐴𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))
 
Theoremeuind 2842* Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)
𝐵 ∈ V    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑦𝐴 = 𝐵)       ((∀𝑥𝑦((𝜑𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧𝑥(𝜑𝑧 = 𝐴))
 
Theoremreu2 2843* A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)
(∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
 
Theoremreu6 2844* A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)
(∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
 
Theoremreu3 2845* A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)
(∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
 
Theoremreu6i 2846* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
 
Theoremeqreu 2847* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝑥 = 𝐵 → (𝜑𝜓))       ((𝐵𝐴𝜓 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
 
Theoremrmo4 2848* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
 
Theoremreu4 2849* Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
 
Theoremreu7 2850* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
 
Theoremreu8 2851* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
 
Theoremrmo3f 2852* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑       (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
 
Theoremrmo4f 2853* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
𝑥𝐴    &   𝑦𝐴    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
 
Theoremreueq 2854* Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
(𝐵𝐴 ↔ ∃!𝑥𝐴 𝑥 = 𝐵)
 
Theoremrmoan 2855 Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 → ∃*𝑥𝐴 (𝜓𝜑))
 
Theoremrmoim 2856 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∀𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
 
Theoremrmoimia 2857 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝑥𝐴 → (𝜑𝜓))       (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑)
 
Theoremrmoimi2 2858 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜓))       (∃*𝑥𝐵 𝜓 → ∃*𝑥𝐴 𝜑)
 
Theorem2reuswapdc 2859* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
(DECID𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑)))
 
Theoremreuind 2860* Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑦𝐴 = 𝐵)       ((∀𝑥𝑦(((𝐴𝐶𝜑) ∧ (𝐵𝐶𝜓)) → 𝐴 = 𝐵) ∧ ∃𝑥(𝐴𝐶𝜑)) → ∃!𝑧𝐶𝑥((𝐴𝐶𝜑) → 𝑧 = 𝐴))
 
Theorem2rmorex 2861* Double restricted quantification with "at most one," analogous to 2moex 2061. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵 ∃*𝑥𝐴 𝜑)
 
Theoremnelrdva 2862* Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑 → ¬ 𝐵𝐴)
 
2.1.7  Conditional equality (experimental)

This is a very useless definition, which "abbreviates" (𝑥 = 𝑦𝜑) as CondEq(𝑥 = 𝑦𝜑). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific ternary operation (𝑥 = 𝑦𝜑).

This is all used as part of a metatheorem: we want to say that (𝑥 = 𝑦 → (𝜑(𝑥) ↔ 𝜑(𝑦))) and (𝑥 = 𝑦𝐴(𝑥) = 𝐴(𝑦)) are provable, for any expressions 𝜑(𝑥) or 𝐴(𝑥) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations.

The metatheorem comes with a disjoint variables condition: every variable in 𝜑(𝑥) is assumed disjoint from 𝑥 except 𝑥 itself. For such a proof by induction, we must consider each of the possible forms of 𝜑(𝑥). If it is a variable other than 𝑥, then we have CondEq(𝑥 = 𝑦𝐴 = 𝐴) or CondEq(𝑥 = 𝑦 → (𝜑𝜑)), which is provable by cdeqth 2867 and reflexivity. Since we are only working with class and wff expressions, it can't be 𝑥 itself in set.mm, but if it was we'd have to also prove CondEq(𝑥 = 𝑦𝑥 = 𝑦) (where set equality is being used on the right).

Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to 𝑥 or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder).

In each of the primitive proofs, we are allowed to assume that 𝑦 is disjoint from 𝜑(𝑥) and vice versa, because this is maintained through the induction. This is how we satisfy the disjoint variable conditions of cdeqab1 2872 and cdeqab 2870.

 
Syntaxwcdeq 2863 Extend wff notation to include conditional equality. This is a technical device used in the proof that is the not-free predicate, and that definitions are conservative as a result.
wff CondEq(𝑥 = 𝑦𝜑)
 
Definitiondf-cdeq 2864 Define conditional equality. All the notation to the left of the is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq𝑥𝑦𝜑. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.)
(CondEq(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))
 
Theoremcdeqi 2865 Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥 = 𝑦𝜑)       CondEq(𝑥 = 𝑦𝜑)
 
Theoremcdeqri 2866 Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦𝜑)       (𝑥 = 𝑦𝜑)
 
Theoremcdeqth 2867 Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝜑       CondEq(𝑥 = 𝑦𝜑)
 
Theoremcdeqnot 2868 Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
 
Theoremcdeqal 2869* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
 
Theoremcdeqab 2870* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})
 
Theoremcdeqal1 2871* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 
Theoremcdeqab1 2872* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → {𝑥𝜑} = {𝑦𝜓})
 
Theoremcdeqim 2873 Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))    &   CondEq(𝑥 = 𝑦 → (𝜒𝜃))       CondEq(𝑥 = 𝑦 → ((𝜑𝜒) ↔ (𝜓𝜃)))
 
Theoremcdeqcv 2874 Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦𝑥 = 𝑦)
 
Theoremcdeqeq 2875 Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦𝐴 = 𝐵)    &   CondEq(𝑥 = 𝑦𝐶 = 𝐷)       CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremcdeqel 2876 Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦𝐴 = 𝐵)    &   CondEq(𝑥 = 𝑦𝐶 = 𝐷)       CondEq(𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
 
Theoremnfcdeq 2877* If we have a conditional equality proof, where 𝜑 is 𝜑(𝑥) and 𝜓 is 𝜑(𝑦), and 𝜑(𝑥) in fact does not have 𝑥 free in it according to , then 𝜑(𝑥) ↔ 𝜑(𝑦) unconditionally. This proves that 𝑥𝜑 is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   CondEq(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑𝜓)
 
Theoremnfccdeq 2878* Variation of nfcdeq 2877 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   CondEq(𝑥 = 𝑦𝐴 = 𝐵)       𝐴 = 𝐵
 
2.1.8  Russell's Paradox
 
Theoremru 2879 Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.

In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as 𝐴 ∈ V, asserted that any collection of sets 𝐴 is a set i.e. belongs to the universe V of all sets. In particular, by substituting {𝑥𝑥𝑥} (the "Russell class") for 𝐴, it asserted {𝑥𝑥𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥𝑥𝑥} ∉ V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system.

In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom asserting that 𝐴 is a set only when it is smaller than some other set 𝐵. The intuitionistic set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla] p. "Axioms of CZF and IZF", which we include as ax-sep 4014. (Contributed by NM, 7-Aug-1994.)

{𝑥𝑥𝑥} ∉ V
 
2.1.9  Proper substitution of classes for sets
 
Syntaxwsbc 2880 Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class 𝐴 for setvar variable 𝑥 in wff 𝜑."
wff [𝐴 / 𝑥]𝜑
 
Definitiondf-sbc 2881 Define the proper substitution of a class for a set.

When 𝐴 is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 2905 for our definition, which always evaluates to true for proper classes.

Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 2882 below). Unfortunately, Quine's definition requires a recursive syntactical breakdown of 𝜑, and it does not seem possible to express it with a single closed formula.

If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 2882, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2881 in the form of sbc8g 2887. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of 𝐴 in every use of this definition) we allow direct reference to df-sbc 2881 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The related definition df-csb defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.)

([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
 
Theoremdfsbcq 2882 This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 2881 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 2883 instead of df-sbc 2881. (dfsbcq2 2883 is needed because unlike Quine we do not overload the df-sb 1719 syntax.) As a consequence of these theorems, we can derive sbc8g 2887, which is a weaker version of df-sbc 2881 that leaves substitution undefined when 𝐴 is a proper class.

However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2887, so we will allow direct use of df-sbc 2881. Proper substiution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.)

(𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
 
Theoremdfsbcq2 2883 This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1719 and substitution for class variables df-sbc 2881. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2882. (Contributed by NM, 31-Dec-2016.)
(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
 
Theoremsbsbc 2884 Show that df-sb 1719 and df-sbc 2881 are equivalent when the class term 𝐴 in df-sbc 2881 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1719 for proofs involving df-sbc 2881. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
 
Theoremsbceq1d 2885 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
(𝜑𝐴 = 𝐵)       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜓))
 
Theoremsbceq1dd 2886 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
(𝜑𝐴 = 𝐵)    &   (𝜑[𝐴 / 𝑥]𝜓)       (𝜑[𝐵 / 𝑥]𝜓)
 
Theoremsbc8g 2887 This is the closest we can get to df-sbc 2881 if we start from dfsbcq 2882 (see its comments) and dfsbcq2 2883. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑}))
 
Theoremsbcex 2888 By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.)
([𝐴 / 𝑥]𝜑𝐴 ∈ V)
 
Theoremsbceq1a 2889 Equality theorem for class substitution. Class version of sbequ12 1727. (Contributed by NM, 26-Sep-2003.)
(𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
 
Theoremsbceq2a 2890 Equality theorem for class substitution. Class version of sbequ12r 1728. (Contributed by NM, 4-Jan-2017.)
(𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑𝜑))
 
Theoremspsbc 2891 Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1731 and rspsbc 2961. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
 
Theoremspsbcd 2892 Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1731 and rspsbc 2961. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝐴𝑉)    &   (𝜑 → ∀𝑥𝜓)       (𝜑[𝐴 / 𝑥]𝜓)
 
Theoremsbcth 2893 A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.)
𝜑       (𝐴𝑉[𝐴 / 𝑥]𝜑)
 
Theoremsbcthdv 2894* Deduction version of sbcth 2893. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(𝜑𝜓)       ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)
 
Theoremsbcid 2895 An identity theorem for substitution. See sbid 1730. (Contributed by Mario Carneiro, 18-Feb-2017.)
([𝑥 / 𝑥]𝜑𝜑)
 
Theoremnfsbc1d 2896 Deduction version of nfsbc1 2897. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)
 
Theoremnfsbc1 2897 Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴       𝑥[𝐴 / 𝑥]𝜑
 
Theoremnfsbc1v 2898* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥[𝐴 / 𝑥]𝜑
 
Theoremnfsbcd 2899 Deduction version of nfsbc 2900. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
 
Theoremnfsbc 2900 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
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