Theorem rabexgfGS 32519

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	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
rabexgfGS	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Proof of Theorem rabexgfGS

Step	Hyp	Ref	Expression
1		nfrab1 3464	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
2		rabexgfGS.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	1, 2	dfssf 3999	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴))
4		rabidim1 3466	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
5	3, 4	mpgbir 1797	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
6		elex 3509	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
7		ssexg 5341	. 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ V) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
8	5, 6, 7	sylancr 586	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  Ⅎwnfc 2893  {crab 3443  Vcvv 3488   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993

This theorem is referenced by:  abrexexd  32529