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Theorem nfun 3653
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 3474 . 2  |-  ( A  u.  B )  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
2 nfun.1 . . . . 5  |-  F/_ x A
32nfcri 2615 . . . 4  |-  F/ x  y  e.  A
4 nfun.2 . . . . 5  |-  F/_ x B
54nfcri 2615 . . . 4  |-  F/ x  y  e.  B
63, 5nfor 1877 . . 3  |-  F/ x
( y  e.  A  \/  y  e.  B
)
76nfab 2626 . 2  |-  F/_ x { y  |  ( y  e.  A  \/  y  e.  B ) }
81, 7nfcxfr 2620 1  |-  F/_ x
( A  u.  B
)
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    e. wcel 1762   {cab 2445   F/_wnfc 2608    u. cun 3467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-un 3474
This theorem is referenced by:  csbun  3850  csbungOLD  3851  nfsuc  4942  nfsup  7900  iuncon  19688  ordtconlem1  27528  esumsplit  27689  measvuni  27811  nfsymdif  29035  bnj958  32952  bnj1000  32953  bnj1408  33046  bnj1446  33055  bnj1447  33056  bnj1448  33057  bnj1466  33063  bnj1467  33064
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