Theorem zfpow 3919

Description: Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Assertion

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

Distinct variable group:   x,y,z

Proof of Theorem zfpow

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pow 3918	. 2 |- E.xA.y(A.w(w e. y -> w e. z) -> y e. x)
2		elequ1 1597	. . . . . . 7 |- (w = x -> (w e. y <-> x e. y))
3		elequ1 1597	. . . . . . 7 |- (w = x -> (w e. z <-> x e. z))
4	2, 3	imbi12d 223	. . . . . 6 |- (w = x -> ((w e. y -> w e. z) <-> (x e. y -> x e. z)))
5	4	cbvalv 1791	. . . . 5 |- (A.w(w e. y -> w e. z) <-> A.x(x e. y -> x e. z))
6	5	imbi1i 227	. . . 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 1356	. . 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 1493	. 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 133	1 |- E.xA.y(A.x(x e. y -> x e. z) -> y e. x)

Syntax hints:   -> wi 4  A.wal 1240  E.wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-13 1401  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  el  3922