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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 2879). (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 2825 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓})) |
3 | abid 2716 | . 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 1537 ∈ wcel 2106 {cab 2712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 |
This theorem is referenced by: eqabri 2883 fvelimab 6981 mapsnend 9075 nosupbnd2 27776 noinfbnd2 27791 fvineqsneu 37394 fvineqsneq 37395 ispridlc 38057 ac6s6 38159 dib1dim 41148 prprspr2 47443 |
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