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| Description: Equality of a class variable and a class abstraction (deduction form of eqabb 2881). (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 2827 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓})) | 
| 3 | abid 2718 | . 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 206 = wceq 1540 ∈ 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 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 | 
| This theorem is referenced by: eqabri 2885 fvelimab 6981 mapsnend 9076 nosupbnd2 27761 noinfbnd2 27776 fvineqsneu 37412 fvineqsneq 37413 ispridlc 38077 ac6s6 38179 dib1dim 41167 prprspr2 47505 | 
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