I got an exercise from my teacher to translate formulas of modal logic with modal operator $\nabla$ into formulas with operators $\Box$ and $\Diamond$.
If the set of possible worlds is $X$, the accessibility relation is $R$ and the semantics of $\nabla$ are as follows:
$x \models \nabla\{\phi_1,...,\phi_n \}$ iff $(\forall y \in X:xRy \to (\exists\phi_k: y \models \phi_k)) \wedge (\forall \phi_k\exists y\in X:xRy \wedge y \models \phi_k)$,
what is the corresponding formula with the same meaning without the $\nabla$ operator?
I think that the result is $\nabla\{\phi_1,\dots,\phi_n\} = \Box(\phi_1\lor\dots\lor\phi_n)\wedge(\Diamond\phi_1\wedge\dots\Diamond\wedge\phi_n)$
Is that correct? Thank you.