Theorem eleq1ab 2743

Description: Extension (in the sense of Remark 3 of the comment of df-clab 2742) of elequ1 2154 from formulas of the form "setvar ∈ setvar" to formulas of the form "setvar ∈ class abstraction". This extension does not require ax-8 2149 contrary to elequ1 2154, but recall from Remark 3 of the comment of df-clab 2742 that it can be considered an extension only because of cvjust 2757, which does require ax-8 2149.

This is an instance of eleq1w 2847 where the containing class is a class abstraction, and contrary to it, it can be proved without df-clel 2839. See also eleq1 2852 for general classes.

The straightforward yet important fact that this statement can be proved from FOL= plus df-clab 2742 (hence without ax-ext 2735, df-cleq 2755 or df-clel 2839) 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 2122	. 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
2		df-clab 2742	. 2 ⊢ (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ [𝑥 / 𝑧]𝜑)
3		df-clab 2742	. 2 ⊢ (𝑦 ∈ {𝑧 ∣ 𝜑} ↔ [𝑦 / 𝑧]𝜑)
4	1, 2, 3	3bitr4g 317	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ 𝑦 ∈ {𝑧 ∣ 𝜑}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  [wsb 2101   ∈ wcel 2147  {cab 2741

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742

This theorem is used by:  cleljustab  2744  ralab2  3658  rexab2  3660