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Theorem List for Intuitionistic Logic Explorer - 2701-2800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempm13.183 2701* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only 𝐴 is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
(𝐴𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴𝑧 = 𝐵)))

Theoremrr19.3v 2702* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)
(∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)

Theoremrr19.28v 2703* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.)
(∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))

Theoremelabgt 2704* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2708.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))

Theoremelabgf 2705 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))

Theoremelabf 2706* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)

Theoremelab 2707* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)

Theoremelabg 2708* 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 2709* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝑉 → (𝐴𝐵𝜓))

Theoremelab2 2710* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝐵𝜓)

Theoremelab4g 2711* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))

Theoremelab3gf 2712 Membership in a class abstraction, with a weaker antecedent than elabgf 2705. (Contributed by NM, 6-Sep-2011.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))

Theoremelab3g 2713* Membership in a class abstraction, with a weaker antecedent than elabg 2708. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))

Theoremelab3 2714* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
(𝜓𝐴 ∈ V)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)

Theoremelrabi 2715* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
(𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)

Theoremelrabf 2716 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 2717* Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2719.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))

Theoremelrab 2718* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))

Theoremelrab3 2719* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))

Theoremelrab2 2720* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐶 = {𝑥𝐵𝜑}       (𝐴𝐶 ↔ (𝐴𝐵𝜓))

Theoremralab 2721* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝜓𝜒))

Theoremralrab 2722* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∀𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∀𝑥𝐴 (𝜓𝜒))

Theoremrexab 2723* 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 2724* 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 2725* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑦(𝜑𝜒))

Theoremralrab2 2726* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑦𝐴 (𝜑𝜒))

Theoremrexab2 2727* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))

Theoremrexrab2 2728* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑦𝐴 (𝜑𝜒))

Theoremabidnf 2729* 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 2730* A deduction theorem for converting the inference 𝑥𝐴 => 𝜑 into a closed theorem. Use nfa1 1448 and nfab 2196 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 2729 is useful. (Contributed by NM, 8-Dec-2006.)
(𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴} → (𝜑𝜓))    &   𝜓       (𝑥𝐴𝜑)

Theoremeqeu 2731* A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑)

Theoremeueq 2732* Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
(𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)

Theoremeueq1 2733* Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
𝐴 ∈ V       ∃!𝑥 𝑥 = 𝐴

Theoremeueq2dc 2734* 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 2735* 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 2736* There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
∃*𝑥 𝑥 = 𝐴

Theoremmoeq3dc 2737* "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 2738* "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)
(∀𝑦∃*𝑥𝜑 → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))

Theoremmosub 2739* "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
∃*𝑥𝜑       ∃*𝑥𝑦(𝑦 = 𝐴𝜑)

Theoremmo2icl 2740* Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
(∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)

Theoremmob2 2741* Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))

Theoremmoi2 2742* Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)
(𝑥 = 𝐴 → (𝜑𝜓))       (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑𝜓)) → 𝑥 = 𝐴)

Theoremmob 2743* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))

Theoremmoi 2744* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓𝜒)) → 𝐴 = 𝐵)

Theoremmorex 2745* Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐵 ∈ V    &   (𝑥 = 𝐵 → (𝜑𝜓))       ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓𝐵𝐴))

Theoremeuxfr2dc 2746* 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 2747* 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 2748* 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 2749* A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)
(∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Theoremreu6 2750* A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)
(∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))

Theoremreu3 2751* A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)
(∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))

Theoremreu6i 2752* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)

Theoremeqreu 2753* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝑥 = 𝐵 → (𝜑𝜓))       ((𝐵𝐴𝜓 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)

Theoremrmo4 2754* 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 2755* Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))

Theoremreu7 2756* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))

Theoremreu8 2757* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))

Theoremreueq 2758* Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
(𝐵𝐴 ↔ ∃!𝑥𝐴 𝑥 = 𝐵)

Theoremrmoan 2759 Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 → ∃*𝑥𝐴 (𝜓𝜑))

Theoremrmoim 2760 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 2761 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 2762 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 2763* 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 2764* 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 2765* Double restricted quantification with "at most one," analogous to 2moex 2000. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵 ∃*𝑥𝐴 𝜑)

Theoremnelrdva 2766* 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 3-ary 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 assumption: 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 2771 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 DV assumptions of cdeqab1 2776 and cdeqab 2774.

Syntaxwcdeq 2767 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 2768 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 2769 Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥 = 𝑦𝜑)       CondEq(𝑥 = 𝑦𝜑)

Theoremcdeqri 2770 Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦𝜑)       (𝑥 = 𝑦𝜑)

Theoremcdeqth 2771 Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝜑       CondEq(𝑥 = 𝑦𝜑)

Theoremcdeqnot 2772 Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))

Theoremcdeqal 2773* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Theoremcdeqab 2774* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})

Theoremcdeqal1 2775* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremcdeqab1 2776* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → {𝑥𝜑} = {𝑦𝜓})

Theoremcdeqim 2777 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 2778 Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦𝑥 = 𝑦)

Theoremcdeqeq 2779 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 2780 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 2781* 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 2782* Variation of nfcdeq 2781 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   CondEq(𝑥 = 𝑦𝐴 = 𝐵)       𝐴 = 𝐵

2.1.8  Russell's Paradox

Theoremru 2783 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 3900. (Contributed by NM, 7-Aug-1994.)

{𝑥𝑥𝑥} ∉ V

2.1.9  Proper substitution of classes for sets

Syntaxwsbc 2784 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 2785 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 2809 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 2786 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 2786, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2785 in the form of sbc8g 2791. 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 2785 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 2786 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 2785 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 2787 instead of df-sbc 2785. (dfsbcq2 2787 is needed because unlike Quine we do not overload the df-sb 1660 syntax.) As a consequence of these theorems, we can derive sbc8g 2791, which is a weaker version of df-sbc 2785 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 2791, so we will allow direct use of df-sbc 2785. 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 2787 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 1660 and substitution for class variables df-sbc 2785. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2786. (Contributed by NM, 31-Dec-2016.)
(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))

Theoremsbsbc 2788 Show that df-sb 1660 and df-sbc 2785 are equivalent when the class term 𝐴 in df-sbc 2785 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1660 for proofs involving df-sbc 2785. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)

Theoremsbceq1d 2789 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 2790 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 2791 This is the closest we can get to df-sbc 2785 if we start from dfsbcq 2786 (see its comments) and dfsbcq2 2787. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑}))

Theoremsbcex 2792 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 2793 Equality theorem for class substitution. Class version of sbequ12 1668. (Contributed by NM, 26-Sep-2003.)
(𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))

Theoremsbceq2a 2794 Equality theorem for class substitution. Class version of sbequ12r 1669. (Contributed by NM, 4-Jan-2017.)
(𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑𝜑))

Theoremspsbc 2795 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 1672 and rspsbc 2865. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))

Theoremspsbcd 2796 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 1672 and rspsbc 2865. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝐴𝑉)    &   (𝜑 → ∀𝑥𝜓)       (𝜑[𝐴 / 𝑥]𝜓)

Theoremsbcth 2797 A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.)
𝜑       (𝐴𝑉[𝐴 / 𝑥]𝜑)

Theoremsbcthdv 2798* Deduction version of sbcth 2797. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(𝜑𝜓)       ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)

Theoremsbcid 2799 An identity theorem for substitution. See sbid 1671. (Contributed by Mario Carneiro, 18-Feb-2017.)
([𝑥 / 𝑥]𝜑𝜑)

Theoremnfsbc1d 2800 Deduction version of nfsbc1 2801. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)

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