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Theorem rabexgfGS 29919
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 3309 . . . 4 𝑥{𝑥𝐴𝜑}
2 rabexgfGS.1 . . . 4 𝑥𝐴
31, 2dfss2f 3812 . . 3 ({𝑥𝐴𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴))
4 rabid 3302 . . . 4 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
54simplbi 493 . . 3 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
63, 5mpgbir 1843 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
7 elex 3414 . 2 (𝐴𝑉𝐴 ∈ V)
8 ssexg 5043 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴 ∈ V) → {𝑥𝐴𝜑} ∈ V)
96, 7, 8sylancr 581 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wnfc 2919  {crab 3094  Vcvv 3398  wss 3792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754  ax-sep 5019
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-in 3799  df-ss 3806
This theorem is referenced by:  abrexexd  29926
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