There seems to be two forms of the conditional statement in predicate logic.
$$\forall x\,(P(x)\Rightarrow Q(x))$$
$$(\forall x\in S)\Rightarrow Q(x)$$
Are these equivalent? I'm a bit confused in the second form because It looks like $Q(x)$ is an open sentence however I know that this is just the universal statement:
$$(\forall x\in S)Q(x)$$
albeit written more explicitly. Are these two statements the same? Is the scope different? If the scope is restricted, or if they are not the same, what is their difference? To me the first says "for every $x$, if $x$ is $P(x)$ then $x$ is a $Q(x)$" and the second says "for every $x$ that is a $P(x)$, aforementioned $x$ is a $Q(x)$". Is more presupposed in this case?