Theorem abeq1i 2896

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
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 2787	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
3	2	abeq2i 2895	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
4	3	bicomi 216	1 ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 198   = wceq 1601   ∈ wcel 2107  {cab 2763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1605  df-ex 1824  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774

This theorem is referenced by: (None)