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Mirrors > Home > ILE Home > Th. List > abeq1i | GIF version |
Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.) |
Ref | Expression |
---|---|
abeqri.1 | ⊢ {𝑥 ∣ 𝜑} = 𝐴 |
Ref | Expression |
---|---|
abeq1i | ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid 2127 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
2 | abeqri.1 | . . 3 ⊢ {𝑥 ∣ 𝜑} = 𝐴 | |
3 | 2 | eleq2i 2206 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ 𝐴) |
4 | 1, 3 | bitr3i 185 | 1 ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∈ wcel 1480 {cab 2125 |
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 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 |
This theorem is referenced by: (None) |
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