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Theorem dfuni2 4215
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 4214 . 2  |-  U. A  =  { x  |  E. y ( x  e.  y  /\  y  e.  A ) }
2 exancom 1716 . . . 4  |-  ( E. y ( x  e.  y  /\  y  e.  A )  <->  E. y
( y  e.  A  /\  x  e.  y
) )
3 df-rex 2779 . . . 4  |-  ( E. y  e.  A  x  e.  y  <->  E. y
( y  e.  A  /\  x  e.  y
) )
42, 3bitr4i 255 . . 3  |-  ( E. y ( x  e.  y  /\  y  e.  A )  <->  E. y  e.  A  x  e.  y )
54abbii 2554 . 2  |-  { x  |  E. y ( x  e.  y  /\  y  e.  A ) }  =  { x  |  E. y  e.  A  x  e.  y }
61, 5eqtri 2449 1  |-  U. A  =  { x  |  E. y  e.  A  x  e.  y }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867   {cab 2405   E.wrex 2774   U.cuni 4213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-rex 2779  df-uni 4214
This theorem is referenced by:  nfuni  4219  nfunid  4220  unieq  4221  uniiun  4346
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