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Theorem rabab 3209
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} = {𝑥𝜑}

Proof of Theorem rabab
StepHypRef Expression
1 df-rab 2916 . 2 {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2 vex 3189 . . . 4 𝑥 ∈ V
32biantrur 527 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43abbii 2736 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
51, 4eqtr4i 2646 1 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  {cab 2607  {crab 2911  Vcvv 3186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-rab 2916  df-v 3188
This theorem is referenced by:  notab  3873  intmin2  4469  euen1  7970  cardf2  8713  hsmex2  9199  imageval  31676  rmxyelqirr  36952  dfrcl2  37444
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