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Theorem rabab 3196
 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 2905 . 2 {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2 vex 3176 . . . 4 𝑥 ∈ V
32biantrur 526 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43abbii 2726 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
51, 4eqtr4i 2635 1 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-rab 2905  df-v 3175 This theorem is referenced by:  notab  3856  intmin2  4439  euen1  7912  cardf2  8652  hsmex2  9138  imageval  31207  rmxyelqirr  36493  dfrcl2  36985
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