Theorem rabab 3468

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 3397	. 2 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2		vex 3441	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 530	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	abbii 2800	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
5	1, 4	eqtr4i 2759	1 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  {crab 3396  Vcvv 3437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439

This theorem is referenced by:  notab  4263  intmin2  4927  euen1  8958  dfttrcl2  9623  cardf2  9845  hsmex2  10333  tz9.1regs  35153  fmla0  35449  fmla0xp  35450  fmla0disjsuc  35465  imageval  35995  dfrcl2  43794