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Theorem nfin 3388
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 3173 . 2  |-  ( A  i^i  B )  =  { y  e.  A  |  y  e.  B }
2 nfin.2 . . . 4  |-  F/_ x B
32nfcri 2426 . . 3  |-  F/ x  y  e.  B
4 nfin.1 . . 3  |-  F/_ x A
53, 4nfrab 2734 . 2  |-  F/_ x { y  e.  A  |  y  e.  B }
61, 5nfcxfr 2429 1  |-  F/_ x
( A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   F/_wnfc 2419   {crab 2560    i^i cin 3164
This theorem is referenced by:  csbing  3389  disjxun  4037  nfres  4973  cp  7577  tskwe  7599  iuncon  17170  ptclsg  17325  limciun  19260  finminlem  26334  stoweidlem57  27909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-in 3172
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