Suppose a map of topological spaces $f$ is a Serre fibration, that is, it has the (right) homotopy lifting property with respect to every inclusion of the form $D^n\to D^n \times I$ as in the following diagram:
I am trying to prove that if $f$ is also a weak equivalence then it has the right lifting property with respect to all border inclusions of the form $S^n\to D^{n+1}$.
In addition, I would want to prove the converse, that if a map has this lifting property then it is a Serre fibration and a weak equivalence.
Here is what I have got (it is not much):
Given a commutative square
by applying the homotopy group functor $\pi_n$ at any basepoint, I get a map $$\mathbb{Z}=\pi_n(S^n) \to \pi_n(X) \to \pi_n (Y)$$ that factors though the trivial group (since $D^{n+1}$ is contractible). And since the second map is an isomorphism, then the first map $\pi_n(S^n)\to \pi_n(X)$ is trivial.
This is not of much help and I would like to know hot to use the homotopy lifting property in combination with this to prove the result.
Help is greatly appreciated.