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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 1830 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓)) |
3 | abeq2 2226 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓)) | |
4 | 2, 3 | sylibr 133 | 1 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1314 = wceq 1316 ∈ wcel 1465 {cab 2103 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 |
This theorem is referenced by: sbab 2244 iftrue 3449 iffalse 3452 iniseg 4881 fncnvima2 5509 isoini 5687 dftpos3 6127 unfiexmid 6774 tgval3 12154 txrest 12372 cnblcld 12631 |
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