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Theorem axpow2 4206
Description: A variant of the Axiom of Power Sets ax-pow 4204 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 4204 . 2  |-  E. y A. z ( A. w
( w  e.  z  ->  w  e.  x
)  ->  z  e.  y )
2 dfss2 3182 . . . . 5  |-  ( z 
C_  x  <->  A. w
( w  e.  z  ->  w  e.  x
) )
32imbi1i 315 . . . 4  |-  ( ( z  C_  x  ->  z  e.  y )  <->  ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y ) )
43albii 1556 . . 3  |-  ( A. z ( z  C_  x  ->  z  e.  y )  <->  A. z ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y ) )
54exbii 1572 . 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 200 1  |-  E. y A. z ( z  C_  x  ->  z  e.  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530   E.wex 1531    e. wcel 1696    C_ wss 3165
This theorem is referenced by:  axpow3  4207  pwex  4209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179
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