Theorem rabex2 4190

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 4189	. 2 |- ( A e. _V -> B e. _V )
5	1, 4	ax-mp 5	1 |- B e. _V

Syntax hints:   = wceq 1373   e. wcel 2176   {crab 2488   _Vcvv 2772

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-sep 4162

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-in 3172  df-ss 3179

This theorem is referenced by:  rab2ex  4191