Theorem rabexd 4235

Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4236. (Contributed by AV, 16-Jul-2019.)

Hypotheses

Ref	Expression
rabexd.1	|- B = { x e. A | ps }
rabexd.2	|- ( ph -> A e. V )

Assertion

Ref	Expression
rabexd	|- ( ph -> B e. _V )

Distinct variable group:   x, A

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

Proof of Theorem rabexd

Step	Hyp	Ref	Expression
1		rabexd.1	. 2 |- B = { x e. A | ps }
2		rabexd.2	. . 3 |- ( ph -> A e. V )
3		rabexg 4233	. . 3 |- ( A e. V -> { x e. A | ps } e. _V )
4	2, 3	syl 14	. 2 |- ( ph -> { x e. A | ps } e. _V )
5	1, 4	eqeltrid 2318	1 |- ( ph -> B e. _V )

Syntax hints:   -> wi 4   = 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:  rabex2  4236  psrbasg  14687  psrelbas  14688  psr0cl  14694  psr0lid  14695  psrnegcl  14696  psrlinv  14697  psrgrp  14698  psr1clfi  14701  mplvalcoe  14703  incistruhgr  15940  clwwlkng  16255