![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2777) of
elequ1 2118 from formulas of the form "setvar ∈ setvar" to formulas of
the form "setvar ∈ class
abstraction". This extension does not
require ax-8 2113 contrary to elequ1 2118, but recall from Remark 3 of the
comment of df-clab 2777 that it can be considered an extension only
because
of cvjust 2793, which does require ax-8 2113.
This is an instance of eleq1w 2872 where the containing class is a class abstraction, and contrary to it, it can be proved without df-clel 2870. See also eleq1 2877 for general classes. The straightforward yet important fact that this statement can be proved from FOL= plus df-clab 2777 (hence without ax-ext 2770, df-cleq 2791 or df-clel 2870) 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 2088 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) | |
2 | df-clab 2777 | . 2 ⊢ (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ [𝑥 / 𝑧]𝜑) | |
3 | df-clab 2777 | . 2 ⊢ (𝑦 ∈ {𝑧 ∣ 𝜑} ↔ [𝑦 / 𝑧]𝜑) | |
4 | 1, 2, 3 | 3bitr4g 317 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ 𝑦 ∈ {𝑧 ∣ 𝜑})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 [wsb 2069 ∈ wcel 2111 {cab 2776 |
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 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 |
This theorem is referenced by: cleljustab 2779 ralab2 3636 rexab2 3639 |
Copyright terms: Public domain | W3C validator |