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Mirrors > Home > ILE Home > Th. List > abbi2dv | GIF version |
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) |
Ref | Expression |
---|---|
abbirdv.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Ref | Expression |
---|---|
abbi2dv | ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbirdv.1 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) | |
2 | 1 | alrimiv 1754 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓)) |
3 | abeq2 2146 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓)) | |
4 | 2, 3 | sylibr 137 | 1 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 = wceq 1243 ∈ wcel 1393 {cab 2026 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 |
This theorem is referenced by: sbab 2164 iftrue 3336 iffalse 3339 iniseg 4697 fncnvima2 5288 isoini 5457 dftpos3 5877 |
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