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Theorem nfuni 3829
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 3825 . 2  |-  U. A  =  { y  |  E. z  e.  A  y  e.  z }
2 nfuni.1 . . . 4  |-  F/_ x A
3 nfv 1538 . . . 4  |-  F/ x  y  e.  z
42, 3nfrexxy 2528 . . 3  |-  F/ x E. z  e.  A  y  e.  z
54nfab 2336 . 2  |-  F/_ x { y  |  E. z  e.  A  y  e.  z }
61, 5nfcxfr 2328 1  |-  F/_ x U. A
Colors of variables: wff set class
Syntax hints:   {cab 2174   F/_wnfc 2318   E.wrex 2468   U.cuni 3823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-rex 2473  df-uni 3824
This theorem is referenced by:  nfiota1  5194  nfrecs  6325  nfsup  7008
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