Theorem nfrabw 2714

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 2519	. 2 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2		nfrabw.2	. . . . 5 |- F/_ x A
3	2	nfcri 2368	. . . 4 |- F/ x y e. A
4		nfrabw.1	. . . 4 |- F/ x ph
5	3, 4	nfan 1613	. . 3 |- F/ x ( y e. A /\ ph )
6	5	nfab 2379	. 2 |- F/_ x { y | ( y e. A /\ ph ) }
7	1, 6	nfcxfr 2371	1 |- F/_ x { y e. A | ph }

Syntax hints:   /\ wa 104   F/wnf 1508   e. wcel 2202   {cab 2217   F/_wnfc 2361   {crab 2514

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519

This theorem is referenced by:  nfdif  3328  nfin  3413  nfse  4438  elfvmptrab1  5741  elovmporab  6221  elovmporab1w  6222  mpoxopoveq  6405  nfsup  7190  caucvgprprlemaddq  7927  ctiunct  13060