Theorem abbi1dv 2928

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
abbi1dv.1	⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴))

Assertion

Ref	Expression
abbi1dv	⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem abbi1dv

Step	Hyp	Ref	Expression
1		abbi1dv.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴))
2	1	bicomd 226	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))
3	2	abbi2dv 2927	. 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
4	3	eqcomd 2804	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870

This theorem is referenced by:  abidnf  3642  csbtt  3845  csbie2g  3868  csb0  4314  csbvarg  4339  iinxsng  4973  predep  6142  enfin2i  9732  fin1a2lem11  9821  hashf1  13811  shftuz  14420  psrbaglefi  20610  vmappw  25701  hdmap1fval  39092  hdmapfval  39123  hgmapfval  39182  rabeqcda  39398