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Theorem zfpow 4193
Description: Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfpow  |-  E. x A. y ( A. x
( x  e.  y  ->  x  e.  z )  ->  y  e.  x )
Distinct variable group:    x, y, z

Proof of Theorem zfpow
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-pow 4192 . 2  |-  E. x A. y ( A. w
( w  e.  y  ->  w  e.  z )  ->  y  e.  x )
2 elequ1 2164 . . . . . . 7  |-  ( w  =  x  ->  (
w  e.  y  <->  x  e.  y ) )
3 elequ1 2164 . . . . . . 7  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
42, 3imbi12d 234 . . . . . 6  |-  ( w  =  x  ->  (
( w  e.  y  ->  w  e.  z )  <->  ( x  e.  y  ->  x  e.  z ) ) )
54cbvalv 1929 . . . . 5  |-  ( A. w ( w  e.  y  ->  w  e.  z )  <->  A. x
( x  e.  y  ->  x  e.  z ) )
65imbi1i 238 . . . 4  |-  ( ( A. w ( w  e.  y  ->  w  e.  z )  ->  y  e.  x )  <->  ( A. x ( x  e.  y  ->  x  e.  z )  ->  y  e.  x ) )
76albii 1481 . . 3  |-  ( A. y ( A. w
( w  e.  y  ->  w  e.  z )  ->  y  e.  x )  <->  A. y
( A. x ( x  e.  y  ->  x  e.  z )  ->  y  e.  x ) )
87exbii 1616 . 2  |-  ( E. x A. y ( A. w ( w  e.  y  ->  w  e.  z )  ->  y  e.  x )  <->  E. x A. y ( A. x
( x  e.  y  ->  x  e.  z )  ->  y  e.  x ) )
91, 8mpbi 145 1  |-  E. x A. y ( A. x
( x  e.  y  ->  x  e.  z )  ->  y  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1362   E.wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-13 2162  ax-pow 4192
This theorem depends on definitions:  df-bi 117  df-nf 1472
This theorem is referenced by:  el  4196
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