Theorem nfrabw 2688

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 2494	. 2 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2		nfrabw.2	. . . . 5 |- F/_ x A
3	2	nfcri 2343	. . . 4 |- F/ x y e. A
4		nfrabw.1	. . . 4 |- F/ x ph
5	3, 4	nfan 1589	. . 3 |- F/ x ( y e. A /\ ph )
6	5	nfab 2354	. 2 |- F/_ x { y | ( y e. A /\ ph ) }
7	1, 6	nfcxfr 2346	1 |- F/_ x { y e. A | ph }

Syntax hints:   /\ wa 104   F/wnf 1484   e. wcel 2177   {cab 2192   F/_wnfc 2336   {crab 2489

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rab 2494

This theorem is referenced by:  nfdif  3298  nfin  3383  nfse  4395  elfvmptrab1  5686  elovmporab  6158  elovmporab1w  6159  mpoxopoveq  6338  nfsup  7108  caucvgprprlemaddq  7836  ctiunct  12881