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Theorem dfuni2 3610
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 3609 . 2  |-  U. A  =  { x  |  E. y ( x  e.  y  /\  y  e.  A ) }
2 exancom 1515 . . . 4  |-  ( E. y ( x  e.  y  /\  y  e.  A )  <->  E. y
( y  e.  A  /\  x  e.  y
) )
3 df-rex 2329 . . . 4  |-  ( E. y  e.  A  x  e.  y  <->  E. y
( y  e.  A  /\  x  e.  y
) )
42, 3bitr4i 180 . . 3  |-  ( E. y ( x  e.  y  /\  y  e.  A )  <->  E. y  e.  A  x  e.  y )
54abbii 2169 . 2  |-  { x  |  E. y ( x  e.  y  /\  y  e.  A ) }  =  { x  |  E. y  e.  A  x  e.  y }
61, 5eqtri 2076 1  |-  U. A  =  { x  |  E. y  e.  A  x  e.  y }
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   E.wrex 2324   U.cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-rex 2329  df-uni 3609
This theorem is referenced by:  nfuni  3614  nfunid  3615  unieq  3617  uniiun  3738
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