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Mirrors > Home > MPE Home > Th. List > eleq1ab | Structured version Visualization version GIF version |
Description: Extension (in the sense
of Remark 3 of the comment of df-clab 2800) of
elequ1 2121 from formulas of the form "setvar ∈ setvar" to formulas of
the form "setvar ∈ class
abstraction". This extension does not
require ax-8 2116 contrary to elequ1 2121, but recall from Remark 3 of the
comment of df-clab 2800 that it can be considered an extension only
because
of cvjust 2816, which does require ax-8 2116.
This is an instance of eleq1w 2895 where the containing class is a class abstraction, and contrary to it, it can be proved without df-clel 2893. See also eleq1 2900 for general classes. The straightforward yet important fact that this statement can be proved from FOL= plus df-clab 2800 (hence without ax-ext 2793, df-cleq 2814 or df-clel 2893) was stressed by Mario Carneiro. (Contributed by BJ, 17-Aug-2023.) |
Ref | Expression |
---|---|
eleq1ab | ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ 𝑦 ∈ {𝑧 ∣ 𝜑})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ 2090 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) | |
2 | df-clab 2800 | . 2 ⊢ (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ [𝑥 / 𝑧]𝜑) | |
3 | df-clab 2800 | . 2 ⊢ (𝑦 ∈ {𝑧 ∣ 𝜑} ↔ [𝑦 / 𝑧]𝜑) | |
4 | 1, 2, 3 | 3bitr4g 316 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ 𝑦 ∈ {𝑧 ∣ 𝜑})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 [wsb 2069 ∈ wcel 2114 {cab 2799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2800 |
This theorem is referenced by: cleljustab 2802 ralab2 3688 rexab2 3691 |
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