Prove that $NP^{NP\cap co-NP} = NP$
Since $A \in \mathsf{NP} \cap \mathsf{coNP}$, there is a witness both for $x \in A$ and for $x \notin A$. An NP machine which wishes to (non-deterministically) find out whether $x \in A$ or not can guess the answer ($x \in A$ or $x \notin A$) and then verify it using a witness.