Theorem rabab 3480

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 3413	. 2 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2		vex 3454	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 540	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	abbii 2827	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
5	1, 4	eqtr4i 2786	1 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  {crab 3412  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452

This theorem is used by:  notab  4259  intmin2  4934  euen1  9057  dfttrcl2  9728  cardf2  10027  hsmex2  10514  tz9.1regs  35858  fmla0  36233  fmla0xp  36234  fmla0disjsuc  36249  mirimanoff  36547  onprcALT  36549  cardprcALT  36551  imageval  36784  dfrcl2  44767