$\forall xP(x) \vee \exists yP(y)$ is true

$\implies$ either $\forall xP(x)$ is true **or** $\exists yP(y)$ is true …..$(i)$

Now, if $P(x)$ is true for all $x$ then $P(x)$ will be true for some $x$ also right ?

i.e. $\exists xP(x)$ is true , put this in eq $(i)$

$\implies$ either $\exists xP(x)$ is true **or** $\exists yP(y)$ is true

$\implies \exists xP(x) \vee \exists yP(y)$ is true

write $x$ in place of $y$

$\implies \exists xP(x) \vee \exists xP(x)$ is true

$\implies \exists xP(x) $ is true

$\therefore$ Option A is correct.