Theorem rabex2 4175

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 4174	. 2 |- ( A e. _V -> B e. _V )
5	1, 4	ax-mp 5	1 |- B e. _V

Syntax hints:   = wceq 1364   e. wcel 2164   {crab 2476   _Vcvv 2760

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-sep 4147

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-in 3159  df-ss 3166

This theorem is referenced by:  rab2ex  4176