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Theorem rabexgfGS 29678
 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 3271 . . . 4 𝑥{𝑥𝐴𝜑}
2 rabexgfGS.1 . . . 4 𝑥𝐴
31, 2dfss2f 3743 . . 3 ({𝑥𝐴𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴))
4 rabid 3264 . . . 4 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
54simplbi 485 . . 3 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
63, 5mpgbir 1874 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
7 elex 3364 . 2 (𝐴𝑉𝐴 ∈ V)
8 ssexg 4939 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴 ∈ V) → {𝑥𝐴𝜑} ∈ V)
96, 7, 8sylancr 575 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2145  Ⅎwnfc 2900  {crab 3065  Vcvv 3351   ⊆ wss 3723 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-in 3730  df-ss 3737 This theorem is referenced by:  abrexexd  29685
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