Logical Equivalence Of Quantified Implications

Question

$\forall x (P(x)\rightarrow Q(x))$ is logically equivalent to $(\exists x P(x)\rightarrow\exists y Q(y))$

The proofs I've seen use logical reasoning to prove the equivalence.
Can someone supply a proof using the laws of boolean algebra?

Answer 1

The two sentences are not logically equivalent. The sentence $\exists x P(x) \longrightarrow \exists y Q(y)$ is true in the natural numbers, if $P(x)$ says $x$ is odd, and $Q(y)$ says that $y$ is prime. The sentence $\forall x(P(x)\longrightarrow Q(x))$ is not.

Additionally, one cannot in general expect tools from Boolean algebra to deal with non-trivial assertions that involve quantification.