I want a third order $\Pi^2_1$-comprehension schema so that $\alpha$ in $$\forall x_1,\ldots,x_k, X_1,\ldots,X_l,\Psi_1,\ldots,\Psi_m\exists \Upsilon\forall Y(\Upsilon Y\Leftrightarrow\forall \Phi\alpha(x_1,\ldots,x_k, X_1,\ldots,X_l,Y,\Psi_1,\ldots,\Psi_m, \Phi))$$ is at most a Boolean combination of $\Pi^1_1$-sets; actually, I only need $\alpha$ to be $\Sigma^1_1$, but a restriction to $\Sigma^1_1$ seems unprincipled here. (Compare to these matters the related question I posed: Can a Boolean Set Algebra be Restricted in the Analytical hierarchy?) How do I, in presence also of the second order $\Pi^1_1$-comprehension schema $$\forall x_1,\ldots, x_n, X_1,\ldots, X_k\exists Y\forall a(a\in Y\iff\beta(a, x_1, \ldots, x_n, X_1, ..., X_k)),$$
best state such a second order restriction as I want upon the third order comprehension principle?
