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Theorem zfun 4117
Description: Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfun xy(x(y x x z) → y x)
Distinct variable group:   x,y,z

Proof of Theorem zfun
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 ax-un 4116 . 2 xy(w(y w w z) → y x)
2 elequ2 1579 . . . . . . 7 (w = x → (y wy x))
3 elequ1 1578 . . . . . . 7 (w = x → (w zx z))
42, 3anbi12d 445 . . . . . 6 (w = x → ((y w w z) ↔ (y x x z)))
54cbvexv 1773 . . . . 5 (w(y w w z) ↔ x(y x x z))
65imbi1i 227 . . . 4 ((w(y w w z) → y x) ↔ (x(y x x z) → y x))
76albii 1335 . . 3 (y(w(y w w z) → y x) ↔ y(x(y x x z) → y x))
87exbii 1474 . 2 (xy(w(y w w z) → y x) ↔ xy(x(y x x z) → y x))
91, 8mpbi 133 1 xy(x(y x x z) → y x)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1224  wex 1358
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-un 4116
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  uniex2  4119  bj-uniex2  7278
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