Theorem nfdif 3065

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 2926	. 2 |- ( A \ B ) = { y e. A | -. y e. B }
2		nfdif.2	. . . . 5 |- F/_ x B
3	2	nfcri 2172	. . . 4 |- F/ x y e. B
4	3	nfn 1548	. . 3 |- F/ x -. y e. B
5		nfdif.1	. . 3 |- F/_ x A
6	4, 5	nfrabxy 2490	. 2 |- F/_ x { y e. A | -. y e. B }
7	1, 6	nfcxfr 2175	1 |- F/_ x ( A \ B )

Syntax hints:   -. wn 3   e. wcel 1393   F/_wnfc 2165   {crab 2310   \ cdif 2914

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-dif 2920

This theorem is referenced by: (None)