Theorem axpow2 2734

Description: A variant of the Axiom of Power Sets ax-pow 2732. 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.

Assertion

Ref	Expression
axpow2	|- E.yA.z(z e. y <-> A.w(w e. z -> w e. x))

Distinct variable group:   x,y,z,w

Proof of Theorem axpow2

Step	Hyp	Ref	Expression
1		ax-pow 2732	. 2 |- E.yA.z(A.w(w e. z -> w e. x) -> z e. y)
2	1	bm1.3ii 2696	1 |- E.yA.z(z e. y <-> A.w(w e. z -> w e. x))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   e. wcel 955  E.wex 977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-12 965  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-sep 2693  ax-pow 2732

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978

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