Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  zfpow Structured version   GIF version

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 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 3918 . 2 xy(w(w yw z) → y x)
2 elequ1 1597 . . . . . . 7 (w = x → (w yx y))
3 elequ1 1597 . . . . . . 7 (w = x → (w zx z))
42, 3imbi12d 223 . . . . . 6 (w = x → ((w yw z) ↔ (x yx z)))
54cbvalv 1791 . . . . 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 1356 . . 3 (y(w(w yw z) → y x) ↔ y(x(x yx z) → y x))
87exbii 1493 . 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 1240  ∃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
 Copyright terms: Public domain W3C validator