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Theorem rabab 3510
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 3434 . 2 {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2 vex 3482 . . . 4 𝑥 ∈ V
32biantrur 530 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43abbii 2807 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
51, 4eqtr4i 2766 1 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  {cab 2712  {crab 3433  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480
This theorem is referenced by:  notab  4320  intmin2  4980  euen1  9066  dfttrcl2  9762  cardf2  9981  hsmex2  10471  fmla0  35367  fmla0xp  35368  fmla0disjsuc  35383  imageval  35912  rmxyelqirrOLD  42899  dfrcl2  43664
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