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Theorem axpow3 4602
Description: A variant of the Axiom of Power Sets ax-pow 4599. For any set  x, there exists a set  y whose members are exactly the subsets of  x i.e. the power set of  x. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
axpow3  |-  E. y A. z ( z  C_  x 
<->  z  e.  y )
Distinct variable group:    x, y, z

Proof of Theorem axpow3
StepHypRef Expression
1 axpow2 4601 . . 3  |-  E. y A. z ( z  C_  x  ->  z  e.  y )
21bm1.3ii 4546 . 2  |-  E. y A. z ( z  e.  y  <->  z  C_  x
)
3 bicom 203 . . . 4  |-  ( ( z  C_  x  <->  z  e.  y )  <->  ( z  e.  y  <->  z  C_  x
) )
43albii 1687 . . 3  |-  ( A. z ( z  C_  x 
<->  z  e.  y )  <->  A. z ( z  e.  y  <->  z  C_  x
) )
54exbii 1712 . 2  |-  ( E. y A. z ( z  C_  x  <->  z  e.  y )  <->  E. y A. z ( z  e.  y  <->  z  C_  x
) )
62, 5mpbir 212 1  |-  E. y A. z ( z  C_  x 
<->  z  e.  y )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   A.wal 1435   E.wex 1659    C_ wss 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-pow 4599
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-in 3443  df-ss 3450
This theorem is referenced by: (None)
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