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Theorem rabexgfGS 32375
Description: Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)
Hypothesis
Ref Expression
rabexgfGS.1 𝑥𝐴
Assertion
Ref Expression
rabexgfGS (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Proof of Theorem rabexgfGS
StepHypRef Expression
1 nfrab1 3438 . . . 4 𝑥{𝑥𝐴𝜑}
2 rabexgfGS.1 . . . 4 𝑥𝐴
31, 2dfssf 3967 . . 3 ({𝑥𝐴𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴))
4 rabidim1 3440 . . 3 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
53, 4mpgbir 1793 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
6 elex 3480 . 2 (𝐴𝑉𝐴 ∈ V)
7 ssexg 5324 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴 ∈ V) → {𝑥𝐴𝜑} ∈ V)
85, 6, 7sylancr 585 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wnfc 2875  {crab 3418  Vcvv 3461  wss 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-rab 3419  df-v 3463  df-in 3951  df-ss 3961
This theorem is referenced by:  abrexexd  32382
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