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Theorem rabexf 41757
Description: Separation Scheme in terms of a restricted class abstraction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rabexf.1 𝑥𝐴
rabexf.2 𝐴𝑉
Assertion
Ref Expression
rabexf {𝑥𝐴𝜑} ∈ V

Proof of Theorem rabexf
StepHypRef Expression
1 rabexf.2 . 2 𝐴𝑉
2 rabexf.1 . . 3 𝑥𝐴
32rabexgf 41640 . 2 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
41, 3ax-mp 5 1 {𝑥𝐴𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2112  wnfc 2939  {crab 3113  Vcvv 3444
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901
This theorem is referenced by:  limsupequzmpt2  42347  liminfequzmpt2  42420
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