Theorem nfdif 3325

Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
nfdif.1	|- F/_ x A
nfdif.2	|- F/_ x B

Assertion

Ref	Expression
nfdif	|- F/_ x ( A \ B )

Proof of Theorem nfdif

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdif2 3205	. 2 |- ( A \ B ) = { y e. A | -. y e. B }
2		nfdif.2	. . . . 5 |- F/_ x B
3	2	nfcri 2366	. . . 4 |- F/ x y e. B
4	3	nfn 1704	. . 3 |- F/ x -. y e. B
5		nfdif.1	. . 3 |- F/_ x A
6	4, 5	nfrabw 2712	. 2 |- F/_ x { y e. A | -. y e. B }
7	1, 6	nfcxfr 2369	1 |- F/_ x ( A \ B )

Syntax hints:   -. wn 3   e. wcel 2200   F/_wnfc 2359   {crab 2512   \ cdif 3194

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-dif 3199

This theorem is referenced by: (None)