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Theorem rabab 3502
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 3433 . 2 {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2 vex 3478 . . . 4 𝑥 ∈ V
32biantrur 531 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43abbii 2802 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
51, 4eqtr4i 2763 1 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  {cab 2709  {crab 3432  Vcvv 3474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476
This theorem is referenced by:  notab  4304  intmin2  4979  euen1  9025  dfttrcl2  9718  cardf2  9937  hsmex2  10427  fmla0  34368  fmla0xp  34369  fmla0disjsuc  34384  imageval  34897  rmxyelqirrOLD  41639  dfrcl2  42415
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