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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 2715) of
     elequ1 2115 from formulas of the form "setvar ∈ setvar" to formulas of
     the form "setvar ∈ class
abstraction".  This extension does not
     require ax-8 2110 contrary to elequ1 2115, but recall from Remark 3 of the
     comment of df-clab 2715 that it can be considered an extension only
because
     of cvjust 2731, which does require ax-8 2110. This is an instance of eleq1w 2824 where the containing class is a class abstraction, and contrary to it, it can be proved without df-clel 2816. See also eleq1 2829 for general classes. The straightforward yet important fact that this statement can be proved from FOL= plus df-clab 2715 (hence without ax-ext 2708, df-cleq 2729 or df-clel 2816) 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 2083 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) | |
| 2 | df-clab 2715 | . 2 ⊢ (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ [𝑥 / 𝑧]𝜑) | |
| 3 | df-clab 2715 | . 2 ⊢ (𝑦 ∈ {𝑧 ∣ 𝜑} ↔ [𝑦 / 𝑧]𝜑) | |
| 4 | 1, 2, 3 | 3bitr4g 314 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ 𝑦 ∈ {𝑧 ∣ 𝜑})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 [wsb 2064 ∈ wcel 2108 {cab 2714 | 
| 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 1910 ax-6 1967 ax-7 2007 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 | 
| This theorem is referenced by: cleljustab 2717 ralab2 3703 rexab2 3705 | 
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