I am reading Fitting and Mendelsohn's first-order modal logic. I have some questions about the $\textit{de re}$ and $\textit{de dicto}$ necessity. Assume that we are working with an S4 system, is $(\star)$ or $(\star\star)$ valid?
$$\Box(\exists xA(x)\rightarrow\exists yB(y))\rightarrow(\exists x\Box A(x)\rightarrow\exists y\Box B(y))\tag{$\star$}$$ $$(\exists xA(x)\rightarrow\exists yB(y))\rightarrow(\exists x\Box A(x)\rightarrow\exists y\Box B(y))\tag{$\star\star$}$$
$(\star)$ is quite similar to the distribution axiom for $\Box$, but in this case, there is a $\textit{de re}$/$\textit{de dicto}$ shift. I don't know how to verify or falsify it.
