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Theorem nfdif 3268
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 3149 . 2 (𝐴𝐵) = {𝑦𝐴 ∣ ¬ 𝑦𝐵}
2 nfdif.2 . . . . 5 𝑥𝐵
32nfcri 2323 . . . 4 𝑥 𝑦𝐵
43nfn 1668 . . 3 𝑥 ¬ 𝑦𝐵
5 nfdif.1 . . 3 𝑥𝐴
64, 5nfrabxy 2668 . 2 𝑥{𝑦𝐴 ∣ ¬ 𝑦𝐵}
71, 6nfcxfr 2326 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 2158  wnfc 2316  {crab 2469  cdif 3138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-dif 3143
This theorem is referenced by: (None)
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