Basic question on Implication Could anyone conceive of any predicates and Universe ( in mathematics, in the world, etc ) where we should use $\exists x ( P(x) \to Q(x) )$, and not necessarily $\forall x ( P(x) \to Q(x) )$ ?
I was thinking of some sittuation where the property of P implies property of Q for at least some individual on the domain, but not for all individuals on the domain. i think its non-existant sittuation ?
I tried to think , and the closest i got was to think that since we know that for some $n$, we have that $n$ is prime, plus $ 2^n-1$ is prime. So, i thought of using $\exists n$ ( $n$ is prime $\to$ $2^n -1$ is prime ) .
Would this be correct to use ?
Then i thought i could use  $\exists n$ ( $n$  is  prime $\land$ $2^n - 1 $ is prime ) instead, and those are not semantic equvialent, so now i'm kinda lost.     

Consider $P(x) \equiv Cube(x)$, $Q(x) \equiv Small(x)$ in a world with a small cube $a$ and a big cube $b$. 
Then $\exists x:P(x) \rightarrow Q(x)$ is true (due to $a$ being small), while $\forall x: P(x) \rightarrow Q(x)$ is false (because $b$ is a cube, but is not small).
