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Mirrors > Home > ILE Home > Th. List > abbi1dv | GIF version |
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) |
Ref | Expression |
---|---|
abbildv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴)) |
Ref | Expression |
---|---|
abbi1dv | ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbildv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴)) | |
2 | 1 | alrimiv 1795 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴)) |
3 | abeq1 2188 | . 2 ⊢ ({𝑥 ∣ 𝜓} = 𝐴 ↔ ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | sylibr 132 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 = wceq 1284 ∈ wcel 1433 {cab 2067 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 |
This theorem is referenced by: abidnf 2760 csbtt 2918 csbvarg 2933 csbie2g 2952 abvor0dc 3269 iinxsng 3751 shftuz 9705 |
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