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Mirrors > Home > ILE Home > Th. List > abeq2d | GIF version |
Description: Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.) |
Ref | Expression |
---|---|
abeqd.1 | ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Ref | Expression |
---|---|
abeq2d | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeqd.1 | . . 3 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) | |
2 | 1 | eleq2d 2245 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓})) |
3 | abid 2163 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓) | |
4 | 2, 3 | bitrdi 196 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2146 {cab 2161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1445 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-ext 2157 |
This theorem depends on definitions: df-bi 117 df-sb 1761 df-clab 2162 df-cleq 2168 df-clel 2171 |
This theorem is referenced by: fvelimab 5564 frecsuclem 6397 |
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