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Theorem nfdif 3094
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 2982 . 2  |-  ( A 
\  B )  =  { y  e.  A  |  -.  y  e.  B }
2 nfdif.2 . . . . 5  |-  F/_ x B
32nfcri 2214 . . . 4  |-  F/ x  y  e.  B
43nfn 1589 . . 3  |-  F/ x  -.  y  e.  B
5 nfdif.1 . . 3  |-  F/_ x A
64, 5nfrabxy 2535 . 2  |-  F/_ x { y  e.  A  |  -.  y  e.  B }
71, 6nfcxfr 2217 1  |-  F/_ x
( A  \  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1434   F/_wnfc 2207   {crab 2353    \ cdif 2971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-dif 2976
This theorem is referenced by: (None)
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