AD implies countable choice for subsets of Baire space

In a natural-number game let I's initial move be $n$, and let II's successive moves form $x\in {\mathbb{N}}^{\mathbb{N}}$. Ignore all later I moves. Declare II the winner exactly when $x\in A_n$, so the complementary condition defines I's payoff as a subset of the full play space. For any particular I strategy its first move is some $n$; nonemptiness of that single $A_n$ gives one $x\in A_n$. Playing its coordinates defeats that strategy regardless of later I moves. Thus no I strategy wins; this argument has made no simultaneous choice from the family.

By A1 the game is determined, and step 1.1 excludes I, so fix a winning II strategy $\tau$. For each $n$ simulate the unique play beginning with I's move $n$ and having all later I moves zero, with II following $\tau$. Recursion on length uniquely defines this play, and Replacement over $n$ forms the sequence of its II subsequences $a_n$. Since every simulated play follows the winning $\tau$, its II subsequence belongs to $A_n$. Hence $(a_n)$ is the promised selection. This includes $n=0$ and singleton $A_n$ without any extra choice. QED.