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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 1826 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴)) |
3 | abeq1 2222 | . 2 ⊢ ({𝑥 ∣ 𝜓} = 𝐴 ↔ ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | sylibr 133 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1310 = wceq 1312 ∈ wcel 1461 {cab 2099 |
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 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 |
This theorem is referenced by: abidnf 2819 csbtt 2979 csbvarg 2994 csbie2g 3014 abvor0dc 3350 iinxsng 3850 shftuz 10476 |
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