Theorem rabexd 4283

Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4284. (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 4281	. . 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 )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821

This proof depends on 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 4251

This proof 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 used by:  rabex2  4284  2omapfi  7326  hashfibclem  11304  cntzval  14153  psrbasg  15178  psrelbas  15179  psrmulrg  15186  psrmulfval  15187  psrmulclfilem  15189  psr0cl  15191  psr0lid  15192  psrnegcl  15193  psrlinv  15194  psrgrp  15195  psr1clfi  15198  psrdifi  15202  psrdirfi  15203  mplvalcoe  15206  incistruhgr  16536  clwwlkng  16851