Question Prove that if an existential formula A is satisfiable in EVERY countable structure, then it's valid.
Proof: My proof is that $B=\lnot A$ is universal so if B is not satisfiable in any countable structure $\iff$ B is not satisfiable in any structure (using Lewenheim Skolem upward theorem) $\iff$ A is valid.
This proof is wrong, as I've saw a hint about using Herbrand's theorem. I want to understand what is wrong with my proof, and how is this proven using Hebrand's theorem.
Herbrand's theorem seems to be overkill. I would say:
Assume that $A$ is true in every finite or countable structure, and that $A$ is not logically valid. We seek a contradiction.
Since $A$ is not logically valid. Then $A$ is false in some structure $M$. Since $A$ is assumed true in every finite structure, $M$ must be infinite. By the downward Löwenheim-Skolem theorem, $M$ is elementarily equivalent to a countable structure, but this is a contradiction because $A$ was assumed to be true in every countable structure.
It is necessary for the truth of the claim that "countable" means "finite or countable". If we require $A$ to be true only in countably infinite structures, then $A\equiv \exists x\exists y(x\ne y)$ is a counterexample -- it is true in every countably infinite structure, but is not logically valid, because it is false in a structure with one element.