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Mirrors > Home > ILE Home > Th. List > abidnf | GIF version |
Description: Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
abidnf | ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 1488 | . . 3 ⊢ (∀𝑥 𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐴) | |
2 | nfcr 2271 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴) | |
3 | 2 | nfrd 1500 | . . 3 ⊢ (Ⅎ𝑥𝐴 → (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)) |
4 | 1, 3 | impbid2 142 | . 2 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥 𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) |
5 | 4 | abbi1dv 2257 | 1 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 = wceq 1331 ∈ wcel 1480 {cab 2123 Ⅎwnfc 2266 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 |
This theorem is referenced by: dedhb 2848 nfopd 3717 nfimad 4885 nffvd 5426 |
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