As shown in Simpson's excellent Subsystems of Second Order Arithmetic, the ‘big five’ system ATR$_0$ from second-order reverse mathematics is equivalent to the following principle:

*For arithmetical $\varphi$ such that $(\forall n)(\exists \text{ at most one } X)\varphi(X, n)$, there is $Z$ such that $(\forall m)(m\in Z\leftrightarrow (\exists X)\varphi(X,m))$.*

My question is whether this ‘at most one’ comprehension principle is also equivalent to the following ‘at most finitely many’ principle, where $w^{1^*}$ is a finite sequence of sets of length $\lvert w\rvert$.

*For arithmetical $\varphi$ such that
$$
(\forall n)(\exists w^{1^*})(\forall X)\Bigl[\varphi(X, n)\rightarrow (\exists i< \lvert w\rvert)(X=w(i))\Bigr],\tag{*}\label{star}
$$
there is $Z$ such that $(\forall m)(m\in Z\leftrightarrow (\exists X)\varphi(X,m))$.*

Note that the condition \eqref{star} guarantees that there are only finitely many $X$ satisfying $\varphi(X,n)$, for fixed $n$.