In (polyadic) first-order logic, for any sentence $\psi(x,y)$ with variables $x$ and $y$ free, we have $(\exists y)(\forall x)\psi(x,y)\vDash (\forall x)(\exists y)\psi(x,y)$ but not the reverse entailment. Is the reverse entailment true in monadic predicate logic, however? And if so, how would a proof go?
It strikes me as a little bizarre that I can't find a reference on this point, but perhaps the answer is obvious and I'm just not seeing it. (Heuristically, though, it would seem to be worth making explicit.)