Theorem rabab 3495

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 3425	. 2 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2		vex 3470	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 530	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	abbii 2794	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
5	1, 4	eqtr4i 2755	1 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701  {crab 3424  Vcvv 3466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468

This theorem is referenced by:  notab  4297  intmin2  4970  euen1  9023  dfttrcl2  9716  cardf2  9935  hsmex2  10425  fmla0  34891  fmla0xp  34892  fmla0disjsuc  34907  imageval  35425  rmxyelqirrOLD  42201  dfrcl2  42975