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Theorem zfun 4237
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 4236 . 2  |-  E. x A. y ( E. w
( y  e.  w  /\  w  e.  z
)  ->  y  e.  x )
2 elequ2 1645 . . . . . . 7  |-  ( w  =  x  ->  (
y  e.  w  <->  y  e.  x ) )
3 elequ1 1644 . . . . . . 7  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
42, 3anbi12d 457 . . . . . 6  |-  ( w  =  x  ->  (
( y  e.  w  /\  w  e.  z
)  <->  ( y  e.  x  /\  x  e.  z ) ) )
54cbvexv 1840 . . . . 5  |-  ( E. w ( y  e.  w  /\  w  e.  z )  <->  E. x
( y  e.  x  /\  x  e.  z
) )
65imbi1i 236 . . . 4  |-  ( ( E. w ( y  e.  w  /\  w  e.  z )  ->  y  e.  x )  <->  ( E. x ( y  e.  x  /\  x  e.  z )  ->  y  e.  x ) )
76albii 1402 . . 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 1539 . 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 143 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 102   A.wal 1285   E.wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-un 4236
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  uniex2  4239  bj-uniex2  11295
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