MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabab Structured version   Visualization version   GIF version

Theorem rabab 3511
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 3436 . 2 {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2 vex 3483 . . . 4 𝑥 ∈ V
32biantrur 530 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43abbii 2808 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
51, 4eqtr4i 2767 1 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wcel 2107  {cab 2713  {crab 3435  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481
This theorem is referenced by:  notab  4313  intmin2  4974  euen1  9068  dfttrcl2  9765  cardf2  9984  hsmex2  10474  fmla0  35388  fmla0xp  35389  fmla0disjsuc  35404  imageval  35932  rmxyelqirrOLD  42927  dfrcl2  43692
  Copyright terms: Public domain W3C validator