pulling universal quantifiers in front of a disjunction

Question

Knowing that $$(∀xP) ∨ (∀xQ)$$ is not equivalent with $$∀x(P ∨ Q),$$ how can I pull the quantifiers in front of the formula simple formula $$ (∀xP) ∨ (∀xQ)$$

Or $$(∃xP) ∧ (∃xQ)$$ for that matter.

Answer 1

The distributive law works here just like it would for finite conjunctions and disjunctions. By analogy with

$$ (w\land x)\lor(y\land z)=(w\lor y)\land(w\lor z)\land(x\lor y)\land(x\lor z) $$

you get

$$ (\forall xP)\lor(\forall xQ)\Leftrightarrow\forall(x,y)(P(x)\lor Q(y))\;, $$

and likewise

$$ (\exists xP)\land(\exists xQ)\Leftrightarrow\exists(x,y)(P(x)\land Q(y))\;. $$

Answer 2

You can change $$(\forall x P(x)) \vee (\forall x Q(x)) $$ to $$(\forall x P(x)) \vee (\forall y Q(y)) $$ by changing the choice of dummy variable, and $\textit{then}$ move the quantifiers to the front to get

$$\forall x \forall y (P(x) \vee Q(y)) $$.