Theorem rabab 2876

Description: A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

Ref	Expression
rabab	⊢ {x ∈ V ∣ φ} = {x ∣ φ}

Proof of Theorem rabab

Step	Hyp	Ref	Expression
1		df-rab 2623	. 2 ⊢ {x ∈ V ∣ φ} = {x ∣ (x ∈ V ∧ φ)}
2		vex 2862	. . . 4 ⊢ x ∈ V
3	2	biantrur 492	. . 3 ⊢ (φ ↔ (x ∈ V ∧ φ))
4	3	abbii 2465	. 2 ⊢ {x ∣ φ} = {x ∣ (x ∈ V ∧ φ)}
5	1, 4	eqtr4i 2376	1 ⊢ {x ∈ V ∣ φ} = {x ∣ φ}

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618  Vcvv 2859

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rab 2623  df-v 2861

This theorem is referenced by:  notab  3525  intmin2  3953