Suppose for two subsets $X\subseteq Y$ of $S$ that there are bases $B_1$ and $B_2$ for which $X\subseteq B_1$ and $B_2\subseteq Y$. Prove that there is a basis $B$ for which $X\subseteq B\subseteq Y$.
I tried to use basis axioms but not succeeded in getting $B$ that satisfies the condition. Any comment is appreciated.
Hint: Start with $B':= B_1 \cap Y$, now apply the base augmentation property to $B'$ and $B_2$ to add new elements to $B'$ until it is a basis.