Theorem nfdif 3302

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 3182	. 2 |- ( A \ B ) = { y e. A | -. y e. B }
2		nfdif.2	. . . . 5 |- F/_ x B
3	2	nfcri 2344	. . . 4 |- F/ x y e. B
4	3	nfn 1682	. . 3 |- F/ x -. y e. B
5		nfdif.1	. . 3 |- F/_ x A
6	4, 5	nfrabw 2689	. 2 |- F/_ x { y e. A | -. y e. B }
7	1, 6	nfcxfr 2347	1 |- F/_ x ( A \ B )

Syntax hints:   -. wn 3   e. wcel 2178   F/_wnfc 2337   {crab 2490   \ cdif 3171

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 615  ax-in2 616  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 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-dif 3176

This theorem is referenced by: (None)