Theorem abidnf 3005

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 1747	. . 3 ⊢ (∀x z ∈ A → z ∈ A)
2		nfcr 2481	. . . 4 ⊢ (ℲxA → Ⅎx z ∈ A)
3	2	nfrd 1763	. . 3 ⊢ (ℲxA → (z ∈ A → ∀x z ∈ A))
4	1, 3	impbid2 195	. 2 ⊢ (ℲxA → (∀x z ∈ A ↔ z ∈ A))
5	4	abbi1dv 2469	1 ⊢ (ℲxA → {z ∣ ∀x z ∈ A} = A)

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478

This theorem is referenced by:  dedhb  3006  nfopd  4605  nfimad  4954  nffvd  5335