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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 1522 | . . 3 ⊢ (∀𝑥 𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐴) | |
2 | nfcr 2324 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴) | |
3 | 2 | nfrd 1531 | . . 3 ⊢ (Ⅎ𝑥𝐴 → (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)) |
4 | 1, 3 | impbid2 143 | . 2 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥 𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) |
5 | 4 | abbi1dv 2309 | 1 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 = wceq 1364 ∈ wcel 2160 {cab 2175 Ⅎwnfc 2319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 |
This theorem is referenced by: dedhb 2921 nfopd 3810 nfimad 4994 nffvd 5542 |
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