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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 2083 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
2 | abeqri.1 | . . 3 ⊢ {𝑥 ∣ 𝜑} = 𝐴 | |
3 | 2 | eleq2i 2161 | . 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 1296 ∈ wcel 1445 {cab 2081 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1388 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 |
This theorem is referenced by: (None) |
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