Theorem abeq1i 2736

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 2631	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
3	2	abeq2i 2735	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
4	3	bicomi 214	1 ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618

This theorem is referenced by: (None)