I'm considering a first-order theory where there is a given binary relation symbol "=" and these axioms (the second one is a axiom schema) : $$\forall x : x = x$$ $$\forall x, y : x=y \rightarrow (F(x) \leftrightarrow F(y))$$ Of course this looks like first-order logic with equality, but I'm looking for examples of models of these axioms where the interpretation of "=" is not the identity relation.
Let's notice that the axioms here imply those of an equivalence relation, but not the converse. What I'm looking for is an interpretation of "=" which is "stricter" than an equivalence relation, but still is not the identity.
I've read some reflexions on the subject (Black's article The Identity of Indiscernibles), but I'd like some mathematical examples, if there are any.