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Theorem axpow3 4415
Description: A variant of the Axiom of Power Sets ax-pow 4412. 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 4414 . . 3  |-  E. y A. z ( z  C_  x  ->  z  e.  y )
21bm1.3ii 4364 . 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 1576 . . 3  |-  ( A. z ( z  C_  x 
<->  z  e.  y )  <->  A. z ( z  e.  y  <->  z  C_  x
) )
54exbii 1593 . 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 1550   E.wex 1551    C_ wss 3309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-pow 4412
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-in 3316  df-ss 3323
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