Theorem rabex2 4236

Description: Separation Scheme in terms of a restricted class abstraction. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)

Hypotheses

Ref	Expression
rabex2.1	|- B = { x e. A | ps }
rabex2.2	|- A e. _V

Assertion

Ref	Expression
rabex2	|- B e. _V

Distinct variable group:   x, A

Allowed substitution hints:   ps( x)   B( x)

Proof of Theorem rabex2

Step	Hyp	Ref	Expression
1		rabex2.2	. 2 |- A e. _V
2		rabex2.1	. . 3 |- B = { x e. A | ps }
3		id 19	. . 3 |- ( A e. _V -> A e. _V )
4	2, 3	rabexd 4235	. 2 |- ( A e. _V -> B e. _V )
5	1, 4	ax-mp 5	1 |- B e. _V

Syntax hints:   = wceq 1397   e. wcel 2202   {crab 2514   _Vcvv 2802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-sep 4207

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-in 3206  df-ss 3213

This theorem is referenced by:  rab2ex  4237