Theorem nfrabxy 2611

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 2425	. 2 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2		nfrabxy.2	. . . . 5 |- F/_ x A
3	2	nfcri 2275	. . . 4 |- F/ x y e. A
4		nfrabxy.1	. . . 4 |- F/ x ph
5	3, 4	nfan 1544	. . 3 |- F/ x ( y e. A /\ ph )
6	5	nfab 2286	. 2 |- F/_ x { y | ( y e. A /\ ph ) }
7	1, 6	nfcxfr 2278	1 |- F/_ x { y e. A | ph }

Syntax hints:   /\ wa 103   F/wnf 1436   e. wcel 1480   {cab 2125   F/_wnfc 2268   {crab 2420

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425

This theorem is referenced by:  nfdif  3197  nfin  3282  nfse  4263  elfvmptrab1  5515  mpoxopoveq  6137  nfsup  6879  caucvgprprlemaddq  7516  ctiunct  11953