MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfiu1 Unicode version

Theorem nfiu1 3933
Description: Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
Assertion
Ref Expression
nfiu1  |-  F/_ x U_ x  e.  A  B

Proof of Theorem nfiu1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-iun 3907 . 2  |-  U_ x  e.  A  B  =  { y  |  E. x  e.  A  y  e.  B }
2 nfre1 2599 . . 3  |-  F/ x E. x  e.  A  y  e.  B
32nfab 2423 . 2  |-  F/_ x { y  |  E. x  e.  A  y  e.  B }
41, 3nfcxfr 2416 1  |-  F/_ x U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   {cab 2269   F/_wnfc 2406   E.wrex 2544   U_ciun 3905
This theorem is referenced by:  ssiun2s  3946  disjxiun  4020  triun  4126  eliunxp  4823  opeliunxp2  4824  ixpf  6838  ixpiunwdom  7305  r1val1  7458  rankuni2b  7525  rankval4  7539  cplem2  7560  ac6num  8106  iunfo  8161  iundom2g  8162  inar1  8397  tskuni  8405  gsum2d2lem  15224  gsum2d2  15225  gsumcom2  15226  iuncon  17154  ptclsg  17309  ssiun2sf  23157  bnj958  28972  bnj1000  28973  bnj981  28982  bnj1398  29064  bnj1408  29066
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-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-rex 2549  df-iun 3907
  Copyright terms: Public domain W3C validator