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.     

$$P(x)\equiv(x\le 1),\qquad Q(x)\equiv(x=1)$$
$$\exists x(x\le1\implies x=1)$$
is true, but
$$\forall x(x\le1\implies x=1)$$
is false.
