Theorem rabid 2788

Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)

Assertion

Ref	Expression
rabid	⊢ (x ∈ {x ∈ A ∣ φ} ↔ (x ∈ A ∧ φ))

Proof of Theorem rabid

Step	Hyp	Ref	Expression
1		df-rab 2624	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
2	1	eqabri 2461	1 ⊢ (x ∈ {x ∈ A ∣ φ} ↔ (x ∈ A ∧ φ))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ 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:  reqabi  2857