Theorem rabab 3487

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 3418	. 2 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2		vex 3461	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 539	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	abbii 2832	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
5	1, 4	eqtr4i 2791	1 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {cab 2743  {crab 3417  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459

This theorem is referenced by:  notab  4269  intmin2  4935  euen1  9012  dfttrcl2  9681  cardf2  9917  hsmex2  10405  tz9.1regs  35437  fmla0  35740  fmla0xp  35741  fmla0disjsuc  35756  imageval  36286  dfrcl2  44257