Theorem abidnf 3699

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 2177	. . 3 ⊢ (∀𝑥 𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐴)
2		nfcr 2889	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴)
3	2	nf5rd 2190	. . 3 ⊢ (Ⅎ𝑥𝐴 → (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
4	1, 3	impbid2 225	. 2 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥 𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
5	4	eqabcdv 2869	1 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2107  {cab 2710  Ⅎwnfc 2884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886

This theorem is referenced by:  dedhb  3700  nfopd  4891  nfimad  6069  nffvd  6904  nfunidALT2  37839  nfunidALT  37840  nfopdALT  37841