Moschovakis proves in Theorem 1C.2 that the finite Borel pointclasses are closed under various operations, like continuous substitution, $\lor,\land,\neg$, etc. He runs the whole proof by induction, assuming that $\bf{\Sigma}^0_n$ are all closed under the operations, and needing to prove that $\Sigma^0_{n+1}$ are closed under them, too.
At the end of the proof, he claims that, assuming closure of $\bf{\Sigma}^0_n$ under $\exists^{\leq}$, then closure of $\bf{\Sigma}^0_{n+1}$ under $\exists^{\leq}$ is trivial. Why is this the case?