Theorem axpow 1082

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(s):   x,y,z

Proof of Theorem axpow

Step	Hyp	Ref	Expression
1		ax-pow 1077	. 2 |- E.xA.y(A.w(w e. y -> w e. z) -> y e. x)
2		a13b 819	. . . . . . 7 |- (w = x -> (w e. y <-> x e. y))
3		a13b 819	. . . . . . 7 |- (w = x -> (w e. z <-> x e. z))
4	2, 3	imbi12d 474	. . . . . 6 |- (w = x -> ((w e. y -> w e. z) <-> (x e. y -> x e. z)))
5	4	cbvalv 972	. . . . 5 |- (A.w(w e. y -> w e. z) <-> A.x(x e. y -> x e. z))
6	5	imbi1i 161	. . . 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	bial 695	. . 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	biex 733	. 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 164	1 |- E.xA.y(A.x(x e. y -> x e. z) -> y e. x)

Syntax hints:   -> wi 2  A.wal 672  E.wex 678   = weq 797   e. wel 803

This theorem is referenced by:  pwex 1806  axpowndlem2 3744

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-13 804  ax-17 925  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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