Theorem abeq1i 2462

Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)

Hypothesis

Ref	Expression
abeqri.1	⊢ {x ∣ φ} = A

Assertion

Ref	Expression
abeq1i	⊢ (φ ↔ x ∈ A)

Proof of Theorem abeq1i

Step	Hyp	Ref	Expression
1		abid 2341	. 2 ⊢ (x ∈ {x ∣ φ} ↔ φ)
2		abeqri.1	. . 3 ⊢ {x ∣ φ} = A
3	2	eleq2i 2417	. 2 ⊢ (x ∈ {x ∣ φ} ↔ x ∈ A)
4	1, 3	bitr3i 242	1 ⊢ (φ ↔ x ∈ A)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349

This theorem is referenced by: (None)