Theorem eleq1ab 2709

Description: Extension (in the sense of Remark 3 of the comment of df-clab 2708) of elequ1 2111 from formulas of the form "setvar ∈ setvar" to formulas of the form "setvar ∈ class abstraction". This extension does not require ax-8 2106 contrary to elequ1 2111, but recall from Remark 3 of the comment of df-clab 2708 that it can be considered an extension only because of cvjust 2724, which does require ax-8 2106.

This is an instance of eleq1w 2814 where the containing class is a class abstraction, and contrary to it, it can be proved without df-clel 2808. See also eleq1 2819 for general classes.

The straightforward yet important fact that this statement can be proved from FOL= plus df-clab 2708 (hence without ax-ext 2701, df-cleq 2722 or df-clel 2808) was stressed by Mario Carneiro. (Contributed by BJ, 17-Aug-2023.)

Assertion

Ref	Expression
eleq1ab	⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ 𝑦 ∈ {𝑧 ∣ 𝜑}))

Proof of Theorem eleq1ab

Step	Hyp	Ref	Expression
1		sbequ 2084	. 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
2		df-clab 2708	. 2 ⊢ (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ [𝑥 / 𝑧]𝜑)
3		df-clab 2708	. 2 ⊢ (𝑦 ∈ {𝑧 ∣ 𝜑} ↔ [𝑦 / 𝑧]𝜑)
4	1, 2, 3	3bitr4g 313	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ 𝑦 ∈ {𝑧 ∣ 𝜑}))

Syntax hints:   → wi 4   ↔ wb 205  [wsb 2065   ∈ wcel 2104  {cab 2707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-sb 2066  df-clab 2708

This theorem is referenced by:  cleljustab  2710  ralab2  3692  rexab2  3694