Theorem eqabcdv 2897

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 226	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))
3	2	eqabdv 2896	. 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
4	3	eqcomd 2769	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {cab 2741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838

This theorem is referenced by:  abidnf  3665  csbtt  3870  csbie2g  3893  csbvarg  4399  iinxsng  5054  predep  6331  fnsnfv  6960  enfin2i  10300  fin1a2lem11  10389  hashf1  14490  shftuz  15102  psrbaglefi  22076  vmappw  27280  addsrid  28157  mulsrid  28306  kard0  35567  hdmap1fval  42570  hdmapfval  42601  hgmapfval  42660