Naively showing that $A_n$ mod a nontrivial normal subgroup is abelian.

Question

Suppose $H \lhd A_n$ is a nontrivial normal subgroup of the alternating group on $n$ letters. Without using the fact that $A_n$ is simple, prove that $A_n/H$ is abelian.

Can this be done?

I will allow use of the fact that the commutator subgroup of $A_n$ is $A_n$.