Theorem rabexgfGS 32003

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 3450	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
2		rabexgfGS.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	1, 2	dfss2f 3973	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴))
4		rabidim1 3452	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
5	3, 4	mpgbir 1800	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
6		elex 3492	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
7		ssexg 5324	. 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 2105  Ⅎwnfc 2882  {crab 3431  Vcvv 3473   ⊆ wss 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-in 3956  df-ss 3966

This theorem is referenced by:  abrexexd  32010