Theorem abidnf 2945

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 1535	. . 3 ⊢ (∀𝑥 𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐴)
2		nfcr 2341	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴)
3	2	nfrd 1544	. . 3 ⊢ (Ⅎ𝑥𝐴 → (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
4	1, 3	impbid2 143	. 2 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥 𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
5	4	abbi1dv 2326	1 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)

Syntax hints:   → wi 4  ∀wal 1371   = wceq 1373   ∈ wcel 2177  {cab 2192  Ⅎwnfc 2336

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338

This theorem is referenced by:  dedhb  2946  nfopd  3845  nfimad  5045  nffvd  5606