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Theorem nfun 3198
Description: Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfun.1  |-  F/_ x A
nfun.2  |-  F/_ x B
Assertion
Ref Expression
nfun  |-  F/_ x
( A  u.  B
)

Proof of Theorem nfun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-un 3041 . 2  |-  ( A  u.  B )  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
2 nfun.1 . . . . 5  |-  F/_ x A
32nfcri 2249 . . . 4  |-  F/ x  y  e.  A
4 nfun.2 . . . . 5  |-  F/_ x B
54nfcri 2249 . . . 4  |-  F/ x  y  e.  B
63, 5nfor 1536 . . 3  |-  F/ x
( y  e.  A  \/  y  e.  B
)
76nfab 2260 . 2  |-  F/_ x { y  |  ( y  e.  A  \/  y  e.  B ) }
81, 7nfcxfr 2252 1  |-  F/_ x
( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \/ wo 680    e. wcel 1463   {cab 2101   F/_wnfc 2242    u. cun 3035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-un 3041
This theorem is referenced by:  nfsuc  4290  nfdju  6879
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