Theorem rabexd 4189

Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4190. (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 4187	. . 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 2292	1 |- ( ph -> B e. _V )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   {crab 2488   _Vcvv 2772

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-sep 4162

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-in 3172  df-ss 3179

This theorem is referenced by:  rabex2  4190  psrbasg  14436  psrelbas  14437  psr0cl  14443  psr0lid  14444  psrnegcl  14445  psrlinv  14446  psrgrp  14447  psr1clfi  14450  mplvalcoe  14452