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Theorem zfun 4225
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 4224 . 2  |-  E. x A. y ( E. w
( y  e.  w  /\  w  e.  z
)  ->  y  e.  x )
2 elequ2 1643 . . . . . . 7  |-  ( w  =  x  ->  (
y  e.  w  <->  y  e.  x ) )
3 elequ1 1642 . . . . . . 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 1838 . . . . 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 1400 . . 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 1537 . 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 1283   E.wex 1422
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-un 4224
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  uniex2  4227  bj-uniex2  11150
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