Theorem nfrabw 2678

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 2484	. 2 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2		nfrabw.2	. . . . 5 |- F/_ x A
3	2	nfcri 2333	. . . 4 |- F/ x y e. A
4		nfrabw.1	. . . 4 |- F/ x ph
5	3, 4	nfan 1579	. . 3 |- F/ x ( y e. A /\ ph )
6	5	nfab 2344	. 2 |- F/_ x { y | ( y e. A /\ ph ) }
7	1, 6	nfcxfr 2336	1 |- F/_ x { y e. A | ph }

Syntax hints:   /\ wa 104   F/wnf 1474   e. wcel 2167   {cab 2182   F/_wnfc 2326   {crab 2479

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484

This theorem is referenced by:  nfdif  3284  nfin  3369  nfse  4376  elfvmptrab1  5656  elovmporab  6123  elovmporab1w  6124  mpoxopoveq  6298  nfsup  7056  caucvgprprlemaddq  7773  ctiunct  12633