Theorem rabab 3466

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 3114	. 2 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2		vex 3440	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 531	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	abbii 2861	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
5	1, 4	eqtr4i 2822	1 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 396   = wceq 1522   ∈ wcel 2081  {cab 2775  {crab 3109  Vcvv 3437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-rab 3114  df-v 3439

This theorem is referenced by:  notab  4193  intmin2  4809  euen1  8427  cardf2  9218  hsmex2  9701  fmla0  32237  fmla0xp  32238  fmla0disjsuc  32253  imageval  33000  rmxyelqirr  38992  dfrcl2  39504