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Theorem rabexgfGS 30189
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 3382 . . . 4 𝑥{𝑥𝐴𝜑}
2 rabexgfGS.1 . . . 4 𝑥𝐴
31, 2dfss2f 3955 . . 3 ({𝑥𝐴𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴))
4 rabidim1 3378 . . 3 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
53, 4mpgbir 1791 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
6 elex 3510 . 2 (𝐴𝑉𝐴 ∈ V)
7 ssexg 5218 . 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 2105  wnfc 2958  {crab 3139  Vcvv 3492  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949
This theorem is referenced by:  abrexexd  30196
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