Distributive properties of quanitifiers What is the difference between
$\forall x~(~P(x) \to Q(x)~)$
and
$\forall x P(x) \to \forall x Q(x)$
To me they seem to be the same thing, what difference does it make where the quantifiers go? 

Suppose some, but not all, $x$ satisfy $P(x)$, and suppose that there is NO $x$ that satisfies $Q(x)$. Then the first statement is false, because there will be some $x$ such that $P(x)$ and not $Q(x)$. The second statement is true, because we have that the premise is false, and the conclusion is false, and any implication with a false premise is by definition, true. Hence the two statements cannot be equivalent.
