MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfun Structured version   Unicode version

Theorem nfun 3495
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 3317 . 2  |-  ( A  u.  B )  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
2 nfun.1 . . . . 5  |-  F/_ x A
32nfcri 2565 . . . 4  |-  F/ x  y  e.  A
4 nfun.2 . . . . 5  |-  F/_ x B
54nfcri 2565 . . . 4  |-  F/ x  y  e.  B
63, 5nfor 1858 . . 3  |-  F/ x
( y  e.  A  \/  y  e.  B
)
76nfab 2575 . 2  |-  F/_ x { y  |  ( y  e.  A  \/  y  e.  B ) }
81, 7nfcxfr 2568 1  |-  F/_ x
( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \/ wo 358    e. wcel 1725   {cab 2421   F/_wnfc 2558    u. cun 3310
This theorem is referenced by:  nfsuc  4644  nfsup  7448  iuncon  17483  esumsplit  24439  measvuni  24560  sbcung  25116  nfsymdif  25659  bnj958  29238  bnj1000  29239  bnj1408  29332  bnj1446  29341  bnj1447  29342  bnj1448  29343  bnj1466  29349  bnj1467  29350
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-un 3317
  Copyright terms: Public domain W3C validator