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Theorem dfuni2 4253
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 4252 . 2  |-  U. A  =  { x  |  E. y ( x  e.  y  /\  y  e.  A ) }
2 exancom 1672 . . . 4  |-  ( E. y ( x  e.  y  /\  y  e.  A )  <->  E. y
( y  e.  A  /\  x  e.  y
) )
3 df-rex 2813 . . . 4  |-  ( E. y  e.  A  x  e.  y  <->  E. y
( y  e.  A  /\  x  e.  y
) )
42, 3bitr4i 252 . . 3  |-  ( E. y ( x  e.  y  /\  y  e.  A )  <->  E. y  e.  A  x  e.  y )
54abbii 2591 . 2  |-  { x  |  E. y ( x  e.  y  /\  y  e.  A ) }  =  { x  |  E. y  e.  A  x  e.  y }
61, 5eqtri 2486 1  |-  U. A  =  { x  |  E. y  e.  A  x  e.  y }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442   E.wrex 2808   U.cuni 4251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-rex 2813  df-uni 4252
This theorem is referenced by:  nfuni  4257  nfunid  4258  unieq  4259  uniiun  4385
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