Proof of the Local Deduction Theorem, for one of many logics

Question

I'm looking for a proof of the Local Deduction Theorem for any one of many logics. That is, a proof of the statement:

$\Sigma \cup \{\phi\} \models \psi$ iff for some positive $n,$ $\Sigma \models \phi^n \rightarrow \psi$

I could use a proof for any fuzzy or related logics, such as MTL, BL, L, Gödel, Product...