Theorem rabeq2i 2857

Description: Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)

Hypothesis

Ref	Expression
rabeqi.1	⊢ A = {x ∈ B ∣ φ}

Assertion

Ref	Expression
rabeq2i	⊢ (x ∈ A ↔ (x ∈ B ∧ φ))

Proof of Theorem rabeq2i

Step	Hyp	Ref	Expression
1		rabeqi.1	. . 3 ⊢ A = {x ∈ B ∣ φ}
2	1	eleq2i 2417	. 2 ⊢ (x ∈ A ↔ x ∈ {x ∈ B ∣ φ})
3		rabid 2788	. 2 ⊢ (x ∈ {x ∈ B ∣ φ} ↔ (x ∈ B ∧ φ))
4	2, 3	bitri 240	1 ⊢ (x ∈ A ↔ (x ∈ B ∧ φ))

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {crab 2619

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  df-rab 2624

This theorem is referenced by:  fvmptss  5706