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Theorem zfpow 3902
Description: Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfpow xy(x(x yx z) → y x)
Distinct variable group:   x,y,z

Proof of Theorem zfpow
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 ax-pow 3901 . 2 xy(w(w yw z) → y x)
2 elequ1 1582 . . . . . . 7 (w = x → (w yx y))
3 elequ1 1582 . . . . . . 7 (w = x → (w zx z))
42, 3imbi12d 223 . . . . . 6 (w = x → ((w yw z) ↔ (x yx z)))
54cbvalv 1776 . . . . 5 (w(w yw z) ↔ x(x yx z))
65imbi1i 227 . . . 4 ((w(w yw z) → y x) ↔ (x(x yx z) → y x))
76albii 1339 . . 3 (y(w(w yw z) → y x) ↔ y(x(x yx z) → y x))
87exbii 1478 . 2 (xy(w(w yw z) → y x) ↔ xy(x(x yx z) → y x))
91, 8mpbi 133 1 xy(x(x yx z) → y x)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226  wex 1362
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-13 1385  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-nf 1330
This theorem is referenced by:  el  3905
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