The strong Spector-Gandy Theorem for the higher analytical pointclasses

Assuming projective determinacy, we extend Spector's strong version of the Spector-Gandy Theorem to all odd levels of the projective hierarchy: Theorem. For every space $X$ which is a finite product of the natural numbers $N$ and Baire space $N^N$ and for every n, if $P$ is a $\Pi^1_{2n+1}$ subset of $X$, then there is a $\Pi^1_{2n}$ set $Q$ such that $P(x) \Longleftrightarrow (\exists!\alpha)Q(x,\alpha) \Longleftrightarrow (\exists\alpha\in\Delta^1_{2n+1}(x))Q(x,\alpha)$.