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Theorem nfin 3282
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 3078 . 2 (𝐴𝐵) = {𝑦𝐴𝑦𝐵}
2 nfin.2 . . . 4 𝑥𝐵
32nfcri 2275 . . 3 𝑥 𝑦𝐵
4 nfin.1 . . 3 𝑥𝐴
53, 4nfrabxy 2611 . 2 𝑥{𝑦𝐴𝑦𝐵}
61, 5nfcxfr 2278 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wcel 1480  wnfc 2268  {crab 2420  cin 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-in 3077
This theorem is referenced by:  csbing  3283  nfres  4821
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