Theorem nfrabxy 2646

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
nfrabxy.1	|- F/ x ph
nfrabxy.2	|- F/_ x A

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem nfrabxy

Step	Hyp	Ref	Expression
1		df-rab 2453	. 2 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2		nfrabxy.2	. . . . 5 |- F/_ x A
3	2	nfcri 2302	. . . 4 |- F/ x y e. A
4		nfrabxy.1	. . . 4 |- F/ x ph
5	3, 4	nfan 1553	. . 3 |- F/ x ( y e. A /\ ph )
6	5	nfab 2313	. 2 |- F/_ x { y | ( y e. A /\ ph ) }
7	1, 6	nfcxfr 2305	1 |- F/_ x { y e. A | ph }

Syntax hints:   /\ wa 103   F/wnf 1448   e. wcel 2136   {cab 2151   F/_wnfc 2295   {crab 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453

This theorem is referenced by:  nfdif  3243  nfin  3328  nfse  4319  elfvmptrab1  5580  mpoxopoveq  6208  nfsup  6957  caucvgprprlemaddq  7649  ctiunct  12373