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Theorem nfdif 4086
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 𝑥𝐴
nfdif.2 𝑥𝐵
Assertion
Ref Expression
nfdif 𝑥(𝐴𝐵)

Proof of Theorem nfdif
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfdif2 3927 . 2 (𝐴𝐵) = {𝑦𝐴 ∣ ¬ 𝑦𝐵}
2 nfdif.2 . . . . 5 𝑥𝐵
32nfcri 2971 . . . 4 𝑥 𝑦𝐵
43nfn 1858 . . 3 𝑥 ¬ 𝑦𝐵
5 nfdif.1 . . 3 𝑥𝐴
64, 5nfrabw 3376 . 2 𝑥{𝑦𝐴 ∣ ¬ 𝑦𝐵}
71, 6nfcxfr 2978 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2115  wnfc 2962  {crab 3136  cdif 3915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-dif 3921
This theorem is referenced by:  nfsymdif  4206  iunxdif3  4998  boxcutc  8488  nfsup  8899  gsum2d2lem  19082  iunconn  22022  iundisj  24141  iundisj2  24142  limciun  24486  difrab2  30256  iundisjf  30336  iundisj2f  30337  suppss2f  30381  aciunf1  30405  iundisjfi  30516  iundisj2fi  30517  fedgmullem2  31047  sigapildsys  31439  csbdif  34644  vvdifopab  35581  compab  40982  iunconnlem2  41477  supminfxr2  41950  stoweidlem28  42512  stoweidlem34  42518  stoweidlem46  42530  stoweidlem53  42537  stoweidlem55  42539  stoweidlem59  42543  stirlinglem5  42562  preimagelt  43179  preimalegt  43180
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