Theorem nfrabw 2733

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 2537	. 2 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2		nfrabw.2	. . . . 5 |- F/_ x A
3	2	nfcri 2386	. . . 4 |- F/ x y e. A
4		nfrabw.1	. . . 4 |- F/ x ph
5	3, 4	nfan 1618	. . 3 |- F/ x ( y e. A /\ ph )
6	5	nfab 2397	. 2 |- F/_ x { y | ( y e. A /\ ph ) }
7	1, 6	nfcxfr 2389	1 |- F/_ x { y e. A | ph }

Syntax hints:   /\ wa 104   F/wnf 1513   e. wcel 2209   {cab 2224   F/_wnfc 2379   {crab 2532

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537

This theorem is referenced by:  nfdif  3350  nfin  3437  nfse  4481  elfvmptrab1  5794  elovmporab  6279  elovmporab1w  6280  mpoxopoveq  6501  nfsup  7322  caucvgprprlemaddq  8065  ctiunct  13309