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Theorem nfun 3503
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 3325 . 2  |-  ( A  u.  B )  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
2 nfun.1 . . . . 5  |-  F/_ x A
32nfcri 2566 . . . 4  |-  F/ x  y  e.  A
4 nfun.2 . . . . 5  |-  F/_ x B
54nfcri 2566 . . . 4  |-  F/ x  y  e.  B
63, 5nfor 1858 . . 3  |-  F/ x
( y  e.  A  \/  y  e.  B
)
76nfab 2576 . 2  |-  F/_ x { y  |  ( y  e.  A  \/  y  e.  B ) }
81, 7nfcxfr 2569 1  |-  F/_ x
( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \/ wo 358    e. wcel 1725   {cab 2422   F/_wnfc 2559    u. cun 3318
This theorem is referenced by:  nfsuc  4652  nfsup  7456  iuncon  17491  esumsplit  24447  measvuni  24568  sbcung  25124  nfsymdif  25667  bnj958  29311  bnj1000  29312  bnj1408  29405  bnj1446  29414  bnj1447  29415  bnj1448  29416  bnj1466  29422  bnj1467  29423
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-un 3325
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