Proof of Homotopy Addition Lemma

Question

Let $X$ be a simplicial set. I'm looking for proof that $[d_0 y] - [d_1 y] + \ldots + (-1)^{n+1}[d_{n+1} y] \in \pi_n(X,x)$ is homotopic to 0, where $y \in X_{n+1}$ and $d_i$ are face maps.

Thanks for your help!