Theorem nfdif 3731

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 3583	. 2 |- ( A \ B ) = { y e. A | -. y e. B }
2		nfdif.2	. . . . 5 |- F/_ x B
3	2	nfcri 2758	. . . 4 |- F/ x y e. B
4	3	nfn 1784	. . 3 |- F/ x -. y e. B
5		nfdif.1	. . 3 |- F/_ x A
6	4, 5	nfrab 3123	. 2 |- F/_ x { y e. A | -. y e. B }
7	1, 6	nfcxfr 2762	1 |- F/_ x ( A \ B )

Syntax hints:   -. wn 3   e. wcel 1990   F/_wnfc 2751   {crab 2916   \ cdif 3571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-dif 3577

This theorem is referenced by:  nfsymdif  3848  iunxdif3  4606  boxcutc  7951  nfsup  8357  gsum2d2lem  18372  iunconn  21231  iundisj  23316  iundisj2  23317  limciun  23658  difrab2  29339  iundisjf  29402  iundisj2f  29403  suppss2f  29439  aciunf1  29463  iundisjfi  29555  iundisj2fi  29556  sigapildsys  30225  csbdif  33171  vvdifopab  34024  compab  38645  iunconnlem2  39171  supminfxr2  39699  stoweidlem28  40245  stoweidlem34  40251  stoweidlem46  40263  stoweidlem53  40270  stoweidlem55  40272  stoweidlem59  40276  stirlinglem5  40295  preimagelt  40912  preimalegt  40913