why is $\forall x (p(x) \implies q(x)) \not\equiv (\forall x p(x)) \implies (\forall x q(x))$ I'm having a hard time wrapping my head around why $\forall x (p(x) \implies q(x)) \not\equiv (\forall x p(x)) \implies (\forall x q(x))$

Consider the domain of real numbers, where $P$ means "equals $\pi$" and $Q$ means "is rational".
Then $\forall x (P(x)\to Q(x))$ means "All real numbers that equal pi are rational", which is blatantly false.
Where as $\forall x~P(x) \to \forall x~Q(x)$ means "If every real number equals pi, then every real number is rational," which is vacuously true.
Thus there is at least one interpretation where the statements are not equivalent.
