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Theorem nfin 3490
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  |-  F/_ x A
nfin.2  |-  F/_ x B
Assertion
Ref Expression
nfin  |-  F/_ x
( A  i^i  B
)

Proof of Theorem nfin
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfin5 3271 . 2  |-  ( A  i^i  B )  =  { y  e.  A  |  y  e.  B }
2 nfin.2 . . . 4  |-  F/_ x B
32nfcri 2517 . . 3  |-  F/ x  y  e.  B
4 nfin.1 . . 3  |-  F/_ x A
53, 4nfrab 2832 . 2  |-  F/_ x { y  e.  A  |  y  e.  B }
61, 5nfcxfr 2520 1  |-  F/_ x
( A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1717   F/_wnfc 2510   {crab 2653    i^i cin 3262
This theorem is referenced by:  csbing  3491  disjxun  4151  nfres  5088  cp  7748  tskwe  7770  iuncon  17412  ptclsg  17568  restmetu  18489  limciun  19648  finminlem  26012  stoweidlem57  27474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-in 3270
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