Prove that $\left [\exists x P(x) \land \forall y \forall z ((P(y) \land P(z)) \rightarrow y=z)\right ] \rightarrow \exists x (P(x) \land \rightarrow \forall y (P(y) \rightarrow y=x)) $
I will lay out our givens:
$\exists x P(x)$
$\forall y \forall z ((P(y) \land P(z) \rightarrow y=z)$
Our goal:
$\exists x P(x)$
$\forall y (P(y) \rightarrow y=x)$
Let $x=x_{0}$ to be some object such that $P(x_{0})$. Therefore $\exists x P(x)$. Now we assume $P(y)$.
From there i dont know how to use the second given to prove the second goal. I feel like i am missing a detail regarding the assumptions that $P(x_{0})$ and $P(y)$. What am i missing in order to connect or justify the second goal with the second given?