Theorem rabexgfGS 32574

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 3419	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
2		rabexgfGS.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	1, 2	dfssf 3924	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴))
4		rabidim1 3421	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
5	3, 4	mpgbir 1800	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
6		elex 3461	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
7		ssexg 5268	. 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ V) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
8	5, 6, 7	sylancr 587	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2113  Ⅎwnfc 2883  {crab 3399  Vcvv 3440   ⊆ wss 3901

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-in 3908  df-ss 3918

This theorem is referenced by:  abrexexd  32584