Theorem nfrabw 2715

Description: A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)

Hypotheses

Ref	Expression
nfrabw.1	|- F/ x ph
nfrabw.2	|- F/_ x A

Assertion

Ref	Expression
nfrabw	|- F/_ x { y e. A | ph }

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   A( x, y)

Proof of Theorem nfrabw

Step	Hyp	Ref	Expression
1		df-rab 2520	. 2 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2		nfrabw.2	. . . . 5 |- F/_ x A
3	2	nfcri 2369	. . . 4 |- F/ x y e. A
4		nfrabw.1	. . . 4 |- F/ x ph
5	3, 4	nfan 1614	. . 3 |- F/ x ( y e. A /\ ph )
6	5	nfab 2380	. 2 |- F/_ x { y | ( y e. A /\ ph ) }
7	1, 6	nfcxfr 2372	1 |- F/_ x { y e. A | ph }

Syntax hints:   /\ wa 104   F/wnf 1509   e. wcel 2202   {cab 2217   F/_wnfc 2362   {crab 2515

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 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520

This theorem is referenced by:  nfdif  3330  nfin  3415  nfse  4444  elfvmptrab1  5750  elovmporab  6232  elovmporab1w  6233  mpoxopoveq  6449  nfsup  7234  caucvgprprlemaddq  7971  ctiunct  13124