∀x (P(x) → Q(x)) and ∀xP(x) → ∀xQ(x)

Section 1.3 #43.
Determine whether ∀x (P(x) → Q(x)) and ∀xP(x) → ∀xQ(x) are logically equivalent. Justify your answer.

Proof: ∀x (P(x) → Q(x))≡ ∀x (¬P(x) ∨ Q(x)) by OULE (p→q). Since the universal quantifier, namely ∀x, cannot be distributed over a disjunction, it is not logically equivalent to ∀xP(x) → ∀xQ(x).