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Theorem nfun 3344
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 3170 . 2  |-  ( A  u.  B )  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
2 nfun.1 . . . . 5  |-  F/_ x A
32nfcri 2426 . . . 4  |-  F/ x  y  e.  A
4 nfun.2 . . . . 5  |-  F/_ x B
54nfcri 2426 . . . 4  |-  F/ x  y  e.  B
63, 5nfor 1782 . . 3  |-  F/ x
( y  e.  A  \/  y  e.  B
)
76nfab 2436 . 2  |-  F/_ x { y  |  ( y  e.  A  \/  y  e.  B ) }
81, 7nfcxfr 2429 1  |-  F/_ x
( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \/ wo 357    e. wcel 1696   {cab 2282   F/_wnfc 2419    u. cun 3163
This theorem is referenced by:  nfsuc  4479  nfsup  7218  iuncon  17170  esumsplit  23446  measvuni  23557  sbcung  24035  nfsymdif  24437  bnj958  29288  bnj1000  29289  bnj1408  29382  bnj1446  29391  bnj1447  29392  bnj1448  29393  bnj1466  29399  bnj1467  29400
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-un 3170
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