Theorem nfdif 3254

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 3135	. 2 |- ( A \ B ) = { y e. A | -. y e. B }
2		nfdif.2	. . . . 5 |- F/_ x B
3	2	nfcri 2311	. . . 4 |- F/ x y e. B
4	3	nfn 1656	. . 3 |- F/ x -. y e. B
5		nfdif.1	. . 3 |- F/_ x A
6	4, 5	nfrabxy 2655	. 2 |- F/_ x { y e. A | -. y e. B }
7	1, 6	nfcxfr 2314	1 |- F/_ x ( A \ B )

Syntax hints:   -. wn 3   e. wcel 2146   F/_wnfc 2304   {crab 2457   \ cdif 3124

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rab 2462  df-dif 3129

This theorem is referenced by: (None)