My textbook introduces shorthand for "exists unique" qualifier $\exists !y [P(y)]$, where full version is: $$\exists y[P(y) \land \forall y'[P(y') \Rightarrow y' = y]]$$
What I can't get is why $\Rightarrow$, not $\Leftarrow \Rightarrow$ used? Saying $\Rightarrow$ means that $P(y')$ could be false, however $y' = y$ or $P(y')$ holds and yet $y' \not = y$. However, as far as I understand, having unique $y$ means that this and only this $y$ would satisfy statement $P$; hence, whenver statement is true, it must be $y$ and vice versa: whenever it is $y$, statement must be true.
Am I missing something?