Necessary condition for the intersection of two submanifolds to be a submanifold

Question

Let $X$ be an $n$-dimensional manifold. How could I prove that for arbitrary submanifolds $M$, $N$ of dimension $n-m$,$n-k$, if $\forall x\in M\cap N$: $\dim( T_xM\cap T_xN) = n-m-k$ then $M\cap N$ is a submanifold?? This is easy when $N$, $M$ are preimages of regular values of some smooth function but what if they are not?