Theorem eqabcdv 2896

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 225	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))
3	2	eqabdv 2895	. 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
4	3	eqcomd 2768	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1560   ∈ wcel 2142  {cab 2740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837

This theorem is referenced by:  abidnf  3665  csbtt  3869  csbie2g  3892  csbvarg  4388  iinxsng  5045  predep  6317  fnsnfv  6946  enfin2i  10278  fin1a2lem11  10367  hashf1  14470  shftuz  15082  psrbaglefi  21978  vmappw  27180  addsrid  28057  mulsrid  28206  hdmap1fval  42420  hdmapfval  42451  hgmapfval  42510