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Theorem zfpow 4002
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 4001 . 2  |-  E. x A. y ( A. w
( w  e.  y  ->  w  e.  z )  ->  y  e.  x )
2 elequ1 1647 . . . . . . 7  |-  ( w  =  x  ->  (
w  e.  y  <->  x  e.  y ) )
3 elequ1 1647 . . . . . . 7  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
42, 3imbi12d 232 . . . . . 6  |-  ( w  =  x  ->  (
( w  e.  y  ->  w  e.  z )  <->  ( x  e.  y  ->  x  e.  z ) ) )
54cbvalv 1842 . . . . 5  |-  ( A. w ( w  e.  y  ->  w  e.  z )  <->  A. x
( x  e.  y  ->  x  e.  z ) )
65imbi1i 236 . . . 4  |-  ( ( A. w ( w  e.  y  ->  w  e.  z )  ->  y  e.  x )  <->  ( A. x ( x  e.  y  ->  x  e.  z )  ->  y  e.  x ) )
76albii 1404 . . 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 1541 . 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 143 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 1287   E.wex 1426
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-13 1449  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  el  4005
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