Theorem axpow2 2739

Description: A variant of the Axiom of Power Sets ax-pow 2737. 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 2737	. 2 |- E.yA.z(A.w(w e. z -> w e. x) -> z e. y)
2	1	bm1.3ii 2701	1 |- E.yA.z(z e. y <-> A.w(w e. z -> w e. x))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   e. wcel 956  E.wex 978

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-sep 2698  ax-pow 2737

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979

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