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In classical propositional logic and in first order logic, if two formulas are logically equivalent then they are substitutable. That is, if we can prove $A \leftrightarrow B$, then we can substitute $B$ for $A$, for arbitrary formulas $A,B$.

(This result is proven in a number of places: Stephen Cole Kleene, Mathematical logic: Replacement Theorem, page 122; Peter Andrews, An introduction to mathematical logic and type theory (1986): Substitutivity of Implication, page 89 and Extended Substitutivity of Implication and Equivalence, page 94;Joseph Shoenfield, Mathematical Logic, Equivalence Theorem, page 34.

Are there interesting logics in which this is not the case? That is, where we have in the logic under consideration that $A$ is logically equivalent to $B$ but also that they are not substitutable for one another.

I should have been clearer. I am not interested in hearing examples from Epistemic logics (since this is one of my areas of research). The same is true of indexicals.

I am more interested, for example, in substructural logics where the property doesn't hold, or many valued logics, for example.

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In most of the many valued logics surveyed by such works as Ackermann (1967), Bolc and Borowik (1991), and Malinowski (1993), this is indeed the case. These logics typically define a conditional using a truth table, and then using the conditional, a biconditional to establish a logical equivalence. However, an examination of the truth tables reveals that these are generally not mathematical equivalence relations: They are not reflexive, symmetric, and transitive. They are thus not sufficient to establish that two formulas have the same truth value.

Once this is observed, it is practically trivial to define a relation that is a true equivalence. In the case of Lukasiewicz 3-valued logic, this can be done by applying the "definite" operator to the biconditional; L(A↔B) Although it's easy enough to do, as far as I can tell it hasn't been done in the published literature.