Theorem eqabcri 2876

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Nov-2019.)

Hypothesis

Ref	Expression
eqabcri.1	⊢ {𝑥 ∣ 𝜑} = 𝐴

Assertion

Ref	Expression
eqabcri	⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

Proof of Theorem eqabcri

Step	Hyp	Ref	Expression
1		eqabcri.1	. . . 4 ⊢ {𝑥 ∣ 𝜑} = 𝐴
2	1	eqcomi 2739	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
3	2	eqabri 2875	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
4	3	bicomi 223	1 ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2104  {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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808

This theorem is referenced by: (None)