Theorem rabexd 4276

Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4277. (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 4274	. . 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 2325	1 |- ( ph -> B e. _V )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-in 3226  df-ss 3233

This theorem is referenced by:  rabex2  4277  2omapfi  7310  hashfibclem  11260  psrbasg  14988  psrelbas  14989  psr0cl  14995  psr0lid  14996  psrnegcl  14997  psrlinv  14998  psrgrp  14999  psr1clfi  15002  mplvalcoe  15004  incistruhgr  16245  clwwlkng  16560