I am working with a particularly forgiving interpretation of $\Box$ in Modal Predicate Logic. $M, w, g \models \Box \varphi$ iff for every $w' \in W$ such that $wRw'$, $M, w', g \models \varphi$ IF $\varphi$ is defined in $w'$. This mostly means that, for any constant $a$ that is nominated in $\varphi$, if $a \notin D_{w''}$, then $w''$ does not count for whether or not $\Box \varphi$ is true at $w$, even if $w'' \in W$ and $wRw''$.
Is the following principle valid on all frames: $(\Box \forall x \varphi) \to (\forall x \Box \varphi)$?
Intuitively, I would say so. However, when I try tu write a proof by induction on the complexity of formulas, starting with the base case of $\varphi$ being atomic, I struggle to prove it. I can prove the base case, but I am at loss for proving anything else.
I believe that my problems resides in the fact that I do not see how to apply the induction hypothesis. Otherwise, if my intuitions are wrong, I can't seem to find a counter-example.
I don't know whether or not $\varphi$ depends on $x$. If that gives two different answers, I would love to know.
