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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 1867 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴)) |
3 | abeq1 2280 | . 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 1346 = wceq 1348 ∈ wcel 2141 {cab 2156 |
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 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 |
This theorem is referenced by: abidnf 2898 csbtt 3061 csbvarg 3077 csbie2g 3099 abvor0dc 3438 iinxsng 3946 shftuz 10781 |
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