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Theorem nfuni 3745
Description: Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
nfuni.1  |-  F/_ x A
Assertion
Ref Expression
nfuni  |-  F/_ x U. A

Proof of Theorem nfuni
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfuni2 3741 . 2  |-  U. A  =  { y  |  E. z  e.  A  y  e.  z }
2 nfuni.1 . . . 4  |-  F/_ x A
3 nfv 1508 . . . 4  |-  F/ x  y  e.  z
42, 3nfrexxy 2472 . . 3  |-  F/ x E. z  e.  A  y  e.  z
54nfab 2286 . 2  |-  F/_ x { y  |  E. z  e.  A  y  e.  z }
61, 5nfcxfr 2278 1  |-  F/_ x U. A
Colors of variables: wff set class
Syntax hints:   {cab 2125   F/_wnfc 2268   E.wrex 2417   U.cuni 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-uni 3740
This theorem is referenced by:  nfiota1  5093  nfrecs  6207  nfsup  6882
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