Theorem abeq1i 2723

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

Hypothesis

Ref	Expression
abeq1i.1	⊢ {𝑥 ∣ 𝜑} = 𝐴

Assertion

Ref	Expression
abeq1i	⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

Proof of Theorem abeq1i

Step	Hyp	Ref	Expression
1		abeq1i.1	. . . 4 ⊢ {𝑥 ∣ 𝜑} = 𝐴
2	1	eqcomi 2619	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
3	2	abeq2i 2722	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
4	3	bicomi 213	1 ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606

This theorem is referenced by: (None)