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Theorem nfdif 3144
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 3029 . 2 (𝐴𝐵) = {𝑦𝐴 ∣ ¬ 𝑦𝐵}
2 nfdif.2 . . . . 5 𝑥𝐵
32nfcri 2234 . . . 4 𝑥 𝑦𝐵
43nfn 1604 . . 3 𝑥 ¬ 𝑦𝐵
5 nfdif.1 . . 3 𝑥𝐴
64, 5nfrabxy 2569 . 2 𝑥{𝑦𝐴 ∣ ¬ 𝑦𝐵}
71, 6nfcxfr 2237 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1448  wnfc 2227  {crab 2379  cdif 3018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rab 2384  df-dif 3023
This theorem is referenced by: (None)
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