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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 2187 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓})) |
3 | abid 2105 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓) | |
4 | 2, 3 | syl6bb 195 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 {cab 2103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 |
This theorem is referenced by: fvelimab 5445 frecsuclem 6271 |
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