Answer
if $\exists x \forall y P(x, y)$ is false, then the given statement is true. Assume that $\exists x \forall y P(x, y)$ is true, we can know that there exists $a , x$ such that $\forall y P(c, y)$ is true and thus $P(c,d)$ is true for every $d$ in the domain. Using existential generalization we then obtain that $\exists x P(x, d)$ for every $d$ in the domain and universal generalization then finally gives us that $\forall y \exists x P(x, y)$ and thus the given statement is also true and this case. This means that the given statement is always true and thus is a Tautology.
Work Step by Step
if $\exists x \forall y P(x, y)$ is false, then the given statement is true. Assume that $\exists x \forall y P(x, y)$ is true, we can know that there exists $a , x$ such that $\forall y P(c, y)$ is true and thus $P(c,d)$ is true for every $d$ in the domain. Using existential generalization we then obtain that $\exists x P(x, d)$ for every $d$ in the domain and universal generalization then finally gives us that $\forall y \exists x P(x, y)$ and thus the given statement is also true and this case. This means that the given statement is always true and thus is a Tautology.
