Theorem abidnf 2760

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.)

Assertion

Ref	Expression
abidnf	⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)

Distinct variable groups:   𝑥,𝑧   𝑧,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem abidnf

Step	Hyp	Ref	Expression
1		sp 1441	. . 3 ⊢ (∀𝑥 𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐴)
2		nfcr 2211	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴)
3	2	nfrd 1453	. . 3 ⊢ (Ⅎ𝑥𝐴 → (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
4	1, 3	impbid2 141	. 2 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥 𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
5	4	abbi1dv 2198	1 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)

Syntax hints:   → wi 4  ∀wal 1282   = wceq 1284   ∈ wcel 1433  {cab 2067  Ⅎwnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208

This theorem is referenced by:  dedhb  2761  nfopd  3587  nfimad  4697  nffvd  5207