Question: Prove/disprove: If the formula $\exists x \alpha$ is valid then there exists closed terms $t_1, t_2$ s.t $\alpha(t_2/x)\vee \alpha(t_2/x)$ is logically valid.
My answer
I think this is false, because a statement can be valid in a signature with only one element in the domain (so there doesn't neccesarily exist 2 closed terms, there's a case where only one exists. I'd like to verify my thoughts on this.