Theorem axpow 2743

Description: Axiom of Power Sets expressed with fewest number of different variables.

Assertion

Ref	Expression
axpow	|- E.xA.y(A.x(x e. y -> x e. z) -> y e. x)

Distinct variable group:   x,y,z

Proof of Theorem axpow

Step	Hyp	Ref	Expression
1		ax-pow 2742	. 2 |- E.xA.y(A.w(w e. y -> w e. z) -> y e. x)
2		elequ1 1136	. . . . . . 7 |- (w = x -> (w e. y <-> x e. y))
3		elequ1 1136	. . . . . . 7 |- (w = x -> (w e. z <-> x e. z))
4	2, 3	imbi12d 626	. . . . . 6 |- (w = x -> ((w e. y -> w e. z) <-> (x e. y -> x e. z)))
5	4	cbvalv 1314	. . . . 5 |- (A.w(w e. y -> w e. z) <-> A.x(x e. y -> x e. z))
6	5	imbi1i 186	. . . 4 |- ((A.w(w e. y -> w e. z) -> y e. x) <-> (A.x(x e. y -> x e. z) -> y e. x))
7	6	albii 999	. . 3 |- (A.y(A.w(w e. y -> w e. z) -> y e. x) <-> A.y(A.x(x e. y -> x e. z) -> y e. x))
8	7	exbii 1051	. 2 |- (E.xA.y(A.w(w e. y -> w e. z) -> y e. x) <-> E.xA.y(A.x(x e. y -> x e. z) -> y e. x))
9	1, 8	mpbi 189	1 |- E.xA.y(A.x(x e. y -> x e. z) -> y e. x)

Syntax hints:   -> wi 3  A.wal 954   = wceq 956   e. wcel 958  E.wex 980

This theorem is referenced by:  pwex 2745  axpowndlem2 4950

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-13 969  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981

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