Theorem eqabcri 2875

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 2740	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
3	2	eqabri 2874	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
4	3	bicomi 224	1 ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111  {cab 2709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806

This theorem is referenced by:  setinds2regs  35127