Theorem eqabcri 2909

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 2775	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
3	2	eqabri 2908	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
4	3	bicomi 227	1 ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2146  {cab 2744

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841

This theorem is used by:  setinds2regs  35568