Given that $$\exists x(\phi\rightarrow\psi)$$ and $$\exists x\phi$$ are valid, where $x$ is the only, possibly, free variable in $\phi,\psi$. Then, does it follow that $$\exists x\psi?$$
It does not seems that this is true, since, when trying to prove it, using Tarski's definition, we see that, due to the existential quantifiers, there should exist two, possibly different, elements $a,b$ in our universe, where $\phi$ and $\phi\rightarrow\psi$ are true, but, since they are not necessarily the same element, $\psi$ may not be true.
So, trying to find a counterexample, we get, evidently, that either $\phi$ should not be valid or $\psi$ should not be true for every interpetation $\mathfrak{A}$.
Choose $\psi\equiv\bot$, where $\bot$ represents falsity. (You can use something like $0=1$ or $P\land \neg P$ if you like.) Then $\exists x.\varphi\to\psi$ is equivalent to $\exists x.\neg\varphi$. So you know that $\exists x.\varphi$ and $\exists x.\neg\varphi$. It is easy to think up predicates that are sometimes true and sometimes false, e.g. $\varphi(x)\equiv (x = 1)$.