Restricted axiom of choice (${\mathsf{AC}}_X(Y)$)

For sets $X$ and $Y$, ${\mathsf{AC}}_X(Y)$ says that every $X$-indexed family $(A_x{)}_{x\in X}$ of nonempty subsets of $Y$ has a choice function $c:X\to Y$ satisfying $c(x)\in A_x$.