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Theorem nfdif 3563
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.
StepHypRef Expression
1 dfdif2 3422 . 2  |-  ( A 
\  B )  =  { y  e.  A  |  -.  y  e.  B }
2 nfdif.2 . . . . 5  |-  F/_ x B
32nfcri 2557 . . . 4  |-  F/ x  y  e.  B
43nfn 1929 . . 3  |-  F/ x  -.  y  e.  B
5 nfdif.1 . . 3  |-  F/_ x A
64, 5nfrab 2988 . 2  |-  F/_ x { y  e.  A  |  -.  y  e.  B }
71, 6nfcxfr 2562 1  |-  F/_ x
( A  \  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1842   F/_wnfc 2550   {crab 2757    \ cdif 3410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-dif 3416
This theorem is referenced by:  nfsymdif  3673  boxcutc  7549  nfsup  7943  gsum2d2lem  17320  iuncon  20219  iundisj  22248  iundisj2  22249  limciun  22588  iunxdif3  27843  iundisjf  27867  iundisj2f  27868  suppss2f  27906  aciunf1  27933  iundisjfi  28035  iundisj2fi  28036  sigapildsys  28596  csbdif  31237  dvtanlemOLD  31417  compab  36178  iunconlem2  36746  stoweidlem28  37159  stoweidlem34  37165  stoweidlem46  37177  stoweidlem53  37184  stoweidlem55  37186  stoweidlem59  37190  stirlinglem5  37209
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