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Theorem zfun 4356
Description: Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfun  |-  E. x A. y ( E. x
( y  e.  x  /\  x  e.  z
)  ->  y  e.  x )
Distinct variable group:    x, y, z

Proof of Theorem zfun
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-un 4355 . 2  |-  E. x A. y ( E. w
( y  e.  w  /\  w  e.  z
)  ->  y  e.  x )
2 elequ2 1691 . . . . . . 7  |-  ( w  =  x  ->  (
y  e.  w  <->  y  e.  x ) )
3 elequ1 1690 . . . . . . 7  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
42, 3anbi12d 464 . . . . . 6  |-  ( w  =  x  ->  (
( y  e.  w  /\  w  e.  z
)  <->  ( y  e.  x  /\  x  e.  z ) ) )
54cbvexv 1890 . . . . 5  |-  ( E. w ( y  e.  w  /\  w  e.  z )  <->  E. x
( y  e.  x  /\  x  e.  z
) )
65imbi1i 237 . . . 4  |-  ( ( E. w ( y  e.  w  /\  w  e.  z )  ->  y  e.  x )  <->  ( E. x ( y  e.  x  /\  x  e.  z )  ->  y  e.  x ) )
76albii 1446 . . 3  |-  ( A. y ( E. w
( y  e.  w  /\  w  e.  z
)  ->  y  e.  x )  <->  A. y
( E. x ( y  e.  x  /\  x  e.  z )  ->  y  e.  x ) )
87exbii 1584 . 2  |-  ( E. x A. y ( E. w ( y  e.  w  /\  w  e.  z )  ->  y  e.  x )  <->  E. x A. y ( E. x
( y  e.  x  /\  x  e.  z
)  ->  y  e.  x ) )
91, 8mpbi 144 1  |-  E. x A. y ( E. x
( y  e.  x  /\  x  e.  z
)  ->  y  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1329   E.wex 1468
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-un 4355
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  uniex2  4358  bj-uniex2  13144
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