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Theorem axpow3 4163
Description: A variant of the Axiom of Power Sets ax-pow 4160. 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 4162 . . 3  |-  E. y A. z ( z  C_  x  ->  z  e.  y )
21bm1.3ii 4118 . 2  |-  E. y A. z ( z  e.  y  <->  z  C_  x
)
3 bicom 193 . . . 4  |-  ( ( z  C_  x  <->  z  e.  y )  <->  ( z  e.  y  <->  z  C_  x
) )
43albii 1554 . . 3  |-  ( A. z ( z  C_  x 
<->  z  e.  y )  <->  A. z ( z  e.  y  <->  z  C_  x
) )
54exbii 1580 . 2  |-  ( E. y A. z ( z  C_  x  <->  z  e.  y )  <->  E. y A. z ( z  e.  y  <->  z  C_  x
) )
62, 5mpbir 202 1  |-  E. y A. z ( z  C_  x 
<->  z  e.  y )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wal 1532   E.wex 1537    e. wcel 1621    C_ wss 3127
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-pow 4160
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-in 3134  df-ss 3141
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