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Mirrors > Home > MPE Home > Th. List > eqabrd | Structured version Visualization version GIF version |
Description: Equality of a class variable and a class abstraction (deduction form of eqabb 2865). (Contributed by NM, 16-Nov-1995.) |
Ref | Expression |
---|---|
eqabrd.1 | ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Ref | Expression |
---|---|
eqabrd | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqabrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) | |
2 | 1 | eleq2d 2811 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓})) |
3 | abid 2705 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓) | |
4 | 2, 3 | bitrdi 287 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {cab 2701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 |
This theorem is referenced by: eqabri 2869 fvelimab 6955 mapsnend 9033 nosupbnd2 27590 noinfbnd2 27605 fvineqsneu 36793 fvineqsneq 36794 ispridlc 37442 ac6s6 37544 dib1dim 40540 prprspr2 46732 |
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