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Theorem nfin 3375
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 3160 . 2  |-  ( A  i^i  B )  =  { y  e.  A  |  y  e.  B }
2 nfin.2 . . . 4  |-  F/_ x B
32nfcri 2413 . . 3  |-  F/ x  y  e.  B
4 nfin.1 . . 3  |-  F/_ x A
53, 4nfrab 2721 . 2  |-  F/_ x { y  e.  A  |  y  e.  B }
61, 5nfcxfr 2416 1  |-  F/_ x
( A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   F/_wnfc 2406   {crab 2547    i^i cin 3151
This theorem is referenced by:  csbing  3376  disjxun  4021  nfres  4957  cp  7561  tskwe  7583  iuncon  17154  ptclsg  17309  limciun  19244  finminlem  26231  stoweidlem57  27806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-in 3159
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