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Theorem rabexgfGS 32507
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 3456 . . . 4 𝑥{𝑥𝐴𝜑}
2 rabexgfGS.1 . . . 4 𝑥𝐴
31, 2dfssf 3973 . . 3 ({𝑥𝐴𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴))
4 rabidim1 3458 . . 3 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
53, 4mpgbir 1799 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
6 elex 3500 . 2 (𝐴𝑉𝐴 ∈ V)
7 ssexg 5321 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴 ∈ V) → {𝑥𝐴𝜑} ∈ V)
85, 6, 7sylancr 587 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wnfc 2889  {crab 3435  Vcvv 3479  wss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rab 3436  df-v 3481  df-in 3957  df-ss 3967
This theorem is referenced by:  abrexexd  32517
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