Consider a First order (FO) formula $\phi = \exists^*\forall^*\exists^* \psi$ where $\psi$ is quantifier free and function free (No n-ary functions, $n \geq 1$) matrix. I am searching for a formula (Example) in the above FO form such that the right most $\exists^*$ cannot be pulled out to the front of the given first oder formula $\phi$. In other words, there is no choice to pull the right most existential quantifier so that that the new formula be in the form of $\phi' = \exists^*\forall^*\psi$.
Consider a formula $\phi = \exists x \forall y \forall z \exists t, P(x,t) \rightarrow Q(y,z)$ and let us pull the right most $\exists t$. So, $\phi' = \exists x \exists t \forall y \forall z , P(x,t) \rightarrow Q(y,z)$. The question is whether $\phi $ and $\phi'$ equivalent? If Yes/No, then why ? What are the conditions in which we can pull the right most existential/universal quantifier so that get aligned in $\exists^*\forall^*\psi$ or $\forall^*\exists^*\psi $ form?