Distributivity of the universal quantifier on the disjunction

Question

I have read that the following holds: $\forall x X \vee \forall xY \Rightarrow \forall x(X \vee Y)$

I'm trying to prove this, but the weapons I have (universal generalisation and instantiation plus the axiomatic distributivity on $\wedge$) do not seem to take me in the right direction. I wonder if I should be looking for some propositional tautology before working on the quantifiers.

The short question is: do you know how to prove this?

Thank you.

Answer 1

Suppose momentarily $\forall x X$. Let $a$ be in the domain of discourse. Then $X(a)$ holds, and thus $X(a)\vee Y(a)$ holds. Then, $\forall x (X\vee Y)$ by universal generalization.

The same reasoning shows that $\forall x Y \implies \forall x (X\vee Y)$.

Now use the fact that $(\phi\rightarrow \xi),(\psi\rightarrow \xi)\vdash (\phi \vee \psi)\rightarrow \xi$.