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Theorem eleq1ab 2804
Description: Extension (in the sense of Remark 3 of the comment of df-clab 2803) of elequ1 2122 from formulas of the form "setvar setvar" to formulas of the form "setvar class abstraction". This extension does not require ax-8 2117 contrary to elequ1 2122, but recall from Remark 3 of the comment of df-clab 2803 that it can be considered an extension only because of cvjust 2819, which does require ax-8 2117.

This is an instance of eleq1w 2898 where the containing class is a class abstraction, and contrary to it, it can be proved without df-clel 2896. See also eleq1 2903 for general classes.

The straightforward yet important fact that this statement can be proved from FOL= plus df-clab 2803 (hence without ax-ext 2796, df-cleq 2817 or df-clel 2896) was stressed by Mario Carneiro. (Contributed by BJ, 17-Aug-2023.)

Assertion
Ref Expression
eleq1ab (𝑥 = 𝑦 → (𝑥 ∈ {𝑧𝜑} ↔ 𝑦 ∈ {𝑧𝜑}))

Proof of Theorem eleq1ab
StepHypRef Expression
1 sbequ 2091 . 2 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
2 df-clab 2803 . 2 (𝑥 ∈ {𝑧𝜑} ↔ [𝑥 / 𝑧]𝜑)
3 df-clab 2803 . 2 (𝑦 ∈ {𝑧𝜑} ↔ [𝑦 / 𝑧]𝜑)
41, 2, 33bitr4g 317 1 (𝑥 = 𝑦 → (𝑥 ∈ {𝑧𝜑} ↔ 𝑦 ∈ {𝑧𝜑}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  [wsb 2070  wcel 2115  {cab 2802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803
This theorem is referenced by:  cleljustab  2805  ralab2  3674  rexab2  3677
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