Theorem nfrabxy 2550

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 2369	. 2 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2		nfrabxy.2	. . . . 5 |- F/_ x A
3	2	nfcri 2223	. . . 4 |- F/ x y e. A
4		nfrabxy.1	. . . 4 |- F/ x ph
5	3, 4	nfan 1503	. . 3 |- F/ x ( y e. A /\ ph )
6	5	nfab 2234	. 2 |- F/_ x { y | ( y e. A /\ ph ) }
7	1, 6	nfcxfr 2226	1 |- F/_ x { y e. A | ph }

Syntax hints:   /\ wa 103   F/wnf 1395   e. wcel 1439   {cab 2075   F/_wnfc 2216   {crab 2364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369

This theorem is referenced by:  nfdif  3124  nfin  3209  nfse  4179  mpt2xopoveq  6021  nfsup  6743  caucvgprprlemaddq  7330