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Theorem nfin 3206
Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfin.1 𝑥𝐴
nfin.2 𝑥𝐵
Assertion
Ref Expression
nfin 𝑥(𝐴𝐵)

Proof of Theorem nfin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfin5 3006 . 2 (𝐴𝐵) = {𝑦𝐴𝑦𝐵}
2 nfin.2 . . . 4 𝑥𝐵
32nfcri 2222 . . 3 𝑥 𝑦𝐵
4 nfin.1 . . 3 𝑥𝐴
53, 4nfrabxy 2547 . 2 𝑥{𝑦𝐴𝑦𝐵}
61, 5nfcxfr 2225 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wcel 1438  wnfc 2215  {crab 2363  cin 2998
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-in 3005
This theorem is referenced by:  csbing  3207  nfres  4711
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