Let $n \geq 0$. Consider a pushout square in the category of simplicial sets as follows:
$$\begin{array} ~ \Lambda^{n+2}_{i} & \stackrel{}{\longrightarrow} & X \\ \downarrow{} & & \downarrow{f} \\ \partial \Delta^{n+2} & \stackrel{}{\longrightarrow} & Y, \end{array} $$
where $\Lambda^{n+2}_{i}$ denotes the $(n+2,i)$-horn and $\partial \Delta^{n+2}$ denotes the boundary of the standard $(n+2)$-simplex.
I need to show that the simplicial map $f \colon X \to Y$ induces an isomorphism $f_{*} \colon \pi_{n}(X,x) \to \pi_{n}(Y,fx)$ for any choice of basepoint $x \in X$. How would I go about proving this?