# First order Language Universal Quantifier Distribution

I've got a quick question about Universal Quantifiers. Given the following:

$$\forall x (p(x) \vee q(x))$$

Can we do this: $$\forall xp(x) \vee \forall xq(x)$$. i.e can we distribute the "for any" to the p and q?

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Let be $A = \forall x (p(x)\lor q(x))$ and $B = \forall x p(x) \lor \forall x q(x)$. It holds that
$$B\Rightarrow A\text{.}$$
The opposite direction may not hold, which can be seen from the answer of user1331281. However, if $q$ does not depend on $x$, then the opposite direction holds also. In that case we have $$A\Leftrightarrow B$$ and this is known as Frobenius law.