Theorem eqabcdv 2862

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)

Hypothesis

Ref	Expression
eqabcdv.1	⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴))

Assertion

Ref	Expression
eqabcdv	⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem eqabcdv

Step	Hyp	Ref	Expression
1		eqabcdv.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴))
2	1	bicomd 223	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))
3	2	eqabdv 2861	. 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
4	3	eqcomd 2735	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2707

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803

This theorem is referenced by:  abidnf  3662  csbtt  3868  csbie2g  3891  csbvarg  4385  iinxsng  5037  predep  6278  fnsnfv  6902  enfin2i  10215  fin1a2lem11  10304  hashf1  14364  shftuz  14976  psrbaglefi  21833  vmappw  27024  addsrid  27876  mulsrid  28021  hdmap1fval  41775  hdmapfval  41806  hgmapfval  41865