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Theorem axpow2 3957
Description: A variant of the Axiom of Power Sets ax-pow 3955 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
axpow2  |-  E. y A. z ( z  C_  x  ->  z  e.  y )
Distinct variable group:    x, y, z

Proof of Theorem axpow2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-pow 3955 . 2  |-  E. y A. z ( A. w
( w  e.  z  ->  w  e.  x
)  ->  z  e.  y )
2 dfss2 2962 . . . . 5  |-  ( z 
C_  x  <->  A. w
( w  e.  z  ->  w  e.  x
) )
32imbi1i 231 . . . 4  |-  ( ( z  C_  x  ->  z  e.  y )  <->  ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y ) )
43albii 1375 . . 3  |-  ( A. z ( z  C_  x  ->  z  e.  y )  <->  A. z ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y ) )
54exbii 1512 . 2  |-  ( E. y A. z ( z  C_  x  ->  z  e.  y )  <->  E. y A. z ( A. w
( w  e.  z  ->  w  e.  x
)  ->  z  e.  y ) )
61, 5mpbir 138 1  |-  E. y A. z ( z  C_  x  ->  z  e.  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257   E.wex 1397    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959
This theorem is referenced by:  axpow3  3958  pwex  3960
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