Theorem abidnf 3697

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 2174	. . 3 ⊢ (∀𝑥 𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐴)
2		nfcr 2886	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴)
3	2	nf5rd 2187	. . 3 ⊢ (Ⅎ𝑥𝐴 → (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
4	1, 3	impbid2 225	. 2 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥 𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
5	4	eqabcdv 2866	1 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)

Syntax hints:   → wi 4  ∀wal 1537   = wceq 1539   ∈ wcel 2104  {cab 2707  Ⅎwnfc 2881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883

This theorem is referenced by:  dedhb  3698  nfopd  4889  nfimad  6067  nffvd  6902  nfunidALT2  38142  nfunidALT  38143  nfopdALT  38144