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Theorem dfuni2 3745
Description: Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
dfuni2  |-  U. A  =  { x  |  E. y  e.  A  x  e.  y }
Distinct variable group:    x, y, A

Proof of Theorem dfuni2
StepHypRef Expression
1 df-uni 3744 . 2  |-  U. A  =  { x  |  E. y ( x  e.  y  /\  y  e.  A ) }
2 exancom 1588 . . . 4  |-  ( E. y ( x  e.  y  /\  y  e.  A )  <->  E. y
( y  e.  A  /\  x  e.  y
) )
3 df-rex 2423 . . . 4  |-  ( E. y  e.  A  x  e.  y  <->  E. y
( y  e.  A  /\  x  e.  y
) )
42, 3bitr4i 186 . . 3  |-  ( E. y ( x  e.  y  /\  y  e.  A )  <->  E. y  e.  A  x  e.  y )
54abbii 2256 . 2  |-  { x  |  E. y ( x  e.  y  /\  y  e.  A ) }  =  { x  |  E. y  e.  A  x  e.  y }
61, 5eqtri 2161 1  |-  U. A  =  { x  |  E. y  e.  A  x  e.  y }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1332   E.wex 1469    e. wcel 1481   {cab 2126   E.wrex 2418   U.cuni 3743
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-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-rex 2423  df-uni 3744
This theorem is referenced by:  nfuni  3749  nfunid  3750  unieq  3752  uniiun  3873
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