Theorem nfrabxy 2658

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 2464	. 2 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2		nfrabxy.2	. . . . 5 |- F/_ x A
3	2	nfcri 2313	. . . 4 |- F/ x y e. A
4		nfrabxy.1	. . . 4 |- F/ x ph
5	3, 4	nfan 1565	. . 3 |- F/ x ( y e. A /\ ph )
6	5	nfab 2324	. 2 |- F/_ x { y | ( y e. A /\ ph ) }
7	1, 6	nfcxfr 2316	1 |- F/_ x { y e. A | ph }

Syntax hints:   /\ wa 104   F/wnf 1460   e. wcel 2148   {cab 2163   F/_wnfc 2306   {crab 2459

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464

This theorem is referenced by:  nfdif  3258  nfin  3343  nfse  4343  elfvmptrab1  5612  mpoxopoveq  6243  nfsup  6993  caucvgprprlemaddq  7709  ctiunct  12443