Theorem rabex 1706

Description: Separation Scheme in terms of a restricted class abstraction.

Hypothesis

Ref	Expression
zfausrab.1	⊢ A ∈ V

Assertion

Ref	Expression
rabex	⊢ {x ∈ A∣φ} ∈ V

Distinct variable group(s):   x,A

Proof of Theorem rabex

Step	Hyp	Ref	Expression
1		zfausrab.1	. 2 ⊢ A ∈ V
2		rabexg 1705	. 2 ⊢ (A ∈ V → {x ∈ A∣φ} ∈ V)
3	1, 2	ax-mp 6	1 ⊢ {x ∈ A∣φ} ∈ V

Syntax hints:   ∈ wcel 1092  {crab 1204  Vcvv 1348

This theorem is referenced by:  canth 2945  inf3lema 3460  aceq6a 3564  ac6lem 3575  kmlem1 3580  zornlem1 3603  subval 4134  divval 4217  flvalt 4623  revalt 4794  imvalt 4795  ocvalt 5161  shsumvalt 5279  pjfn 5586

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208  df-v 1349  df-in 1491  df-ss 1492

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