What is the reason why the following expression does not simplify to True?

Assuming[
  i > 0 && j > 0 && k > 0 && Element[{i, j, k}, Integers],
  FullSimplify[
    KroneckerDelta[i, j] * KroneckerDelta[i, k] == KroneckerDelta[i, j, k]
  ]
]

(* KroneckerDelta[i, j] KroneckerDelta[i, k] == KroneckerDelta[i, j, k] *)

Is there a corner case where this equation is actually false? Does Mathematica simply not know about this identity? If the latter is the case, how could I define a rule for FullSimplify to simplify products like the one on the left hand side?