How does $P(x_{0}) \land \forall y \forall z ((P(y) \land P(z) \rightarrow y=z) \land P(y)$ imply $y=x_{0}$

Question

Ive been looking at this for a while and have absolutely no clue how these assumptions imply that $y=x_{0}$

Answer 1

Stuff you have:

1. $P(x_0)$
2. $P(y)$
3. $\forall y,z.\ P(y)\land P(z) \rightarrow y=z$

The third one is for all $y$ and $z$, so in particular for the $y$ from before and for $x_0$ respectively. This gives: $$P(y)\land P(x_0) \rightarrow y=x_0$$ As you have both $P(y)$ and $P(x_0)$, you get $y=x_0$, as you desired.