Theorem abeq2i 2461

Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)

Hypothesis

Ref	Expression
abeqi.1	⊢ A = {x ∣ φ}

Assertion

Ref	Expression
abeq2i	⊢ (x ∈ A ↔ φ)

Proof of Theorem abeq2i

Step	Hyp	Ref	Expression
1		abeqi.1	. . 3 ⊢ A = {x ∣ φ}
2	1	eleq2i 2417	. 2 ⊢ (x ∈ A ↔ x ∈ {x ∣ φ})
3		abid 2341	. 2 ⊢ (x ∈ {x ∣ φ} ↔ φ)
4	2, 3	bitri 240	1 ⊢ (x ∈ A ↔ φ)

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:  rabid  2788  vex  2863  csbco  3146  csbnestgf  3185  pwss  3737  elsn  3749  snsspw  3878  1cex  4143  elp6  4264  unipw1  4326  phi011lem1  4599  fconstopab  4816  fvfullfunlem3  5864  fvfullfun  5865  pmvalg  6011  enpw1pw  6076