I will answer in a way that will likely add confusion, but I hope shows why this type of question should not be seen as too obvious and easy. I apologise in advance.
One question that comes up is "what does it mean to assert $\forall x (P(x))$ when there are infinite x?" Why do students ask such a silly question? Because one way to answer it is to say "it is a fact that all x obey P" or "it is known that all x obey P" or "it has been shown P(x) for all x" or "for every x we can prove P(x)". Something like that is what a lot of students have in mind, and when one is dealing with infinities, things like knowing or proving may never get finished.
Similarly, we can ask what it means to show $\exists x (P(x))$. One way that students will often think about this is that showing that is giving a particular x that obeys P.
Now, if these are the things the student is thinking, then it certainly seems reasonable to ask the question of the OP. Even if we can assert that "it is known that not all x obey P" or worse, that "for every x we cannot prove P(x)", this does not mean we have given a particular x that obeys P! So how can we assert existence here? What if we just don't know?
Of course, the answer is that there are different, completely valid ways to answer this. On one side, you have the constructivists who will smile at these questions, pat you on the shoulder, and tell you "good questions, all" as they lead you off on explanations of the meaning of truth and accessible knowledge. On another side, you have the classicists, who will try to help this poor student understand that there is much more to truth than knowledge and proof, and try to build in that student reasons why bivalent reasoning about these kinds of things is still important and how can reach assertions of existence nonconstructively.
These are both valid paths one may walk down in the philosophy of mathematics. They offer different ways to understand the meaning of assertions, and it often seems like both sides are talking past each other when they are often just talking about different things entirely. A curious student might want to get familiar with both sides, and learn the ways each approach formal manipulation.
A noncurious student, though, should probably just learn the rule and move on. Beyond here, there be dragons.