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Theorem nfun 3331
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 3157 . 2  |-  ( A  u.  B )  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
2 nfun.1 . . . . 5  |-  F/_ x A
32nfcri 2413 . . . 4  |-  F/ x  y  e.  A
4 nfun.2 . . . . 5  |-  F/_ x B
54nfcri 2413 . . . 4  |-  F/ x  y  e.  B
63, 5nfor 1770 . . 3  |-  F/ x
( y  e.  A  \/  y  e.  B
)
76nfab 2423 . 2  |-  F/_ x { y  |  ( y  e.  A  \/  y  e.  B ) }
81, 7nfcxfr 2416 1  |-  F/_ x
( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \/ wo 357    e. wcel 1684   {cab 2269   F/_wnfc 2406    u. cun 3150
This theorem is referenced by:  nfsuc  4463  nfsup  7202  iuncon  17154  esumsplit  23431  measvuni  23542  sbcung  24020  nfsymdif  24366  bnj958  28972  bnj1000  28973  bnj1408  29066  bnj1446  29075  bnj1447  29076  bnj1448  29077  bnj1466  29083  bnj1467  29084
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-un 3157
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