Theorem rabexd 4262

Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4263. (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 4260	. . 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 2321	1 |- ( ph -> B e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   {crab 2526   _Vcvv 2815

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-sep 4233

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-v 2817  df-in 3220  df-ss 3227

This theorem is referenced by:  rabex2  4263  2omapfi  7284  hashfibclem  11231  psrbasg  14955  psrelbas  14956  psr0cl  14962  psr0lid  14963  psrnegcl  14964  psrlinv  14965  psrgrp  14966  psr1clfi  14969  mplvalcoe  14971  incistruhgr  16211  clwwlkng  16526