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Theorem nfin 3539
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 3320 . 2  |-  ( A  i^i  B )  =  { y  e.  A  |  y  e.  B }
2 nfin.2 . . . 4  |-  F/_ x B
32nfcri 2565 . . 3  |-  F/ x  y  e.  B
4 nfin.1 . . 3  |-  F/_ x A
53, 4nfrab 2881 . 2  |-  F/_ x { y  e.  A  |  y  e.  B }
61, 5nfcxfr 2568 1  |-  F/_ x
( A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   F/_wnfc 2558   {crab 2701    i^i cin 3311
This theorem is referenced by:  csbing  3540  disjxun  4202  nfres  5140  cp  7807  tskwe  7829  iuncon  17483  ptclsg  17639  restmetu  18609  limciun  19773  nfpred  25436  mbfposadd  26244  finminlem  26312  stoweidlem57  27773  iunconlem2  28984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-in 3319
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