I have to prove the following statement: $(xy)^{-1}=x^{-1}y^{-1}$ $\forall x,y\in G$ if and only if $G$ is Abelian. $G$ is a group and $x,y\in G$. This is what I did:

If $G$ is Abelian, then $(xy)^{-1}=y^{-1}x^{-1}=x^{-1}y^{-1}$ for all $x,y\in G$.
Now suppose that $(xy)^{-1}=x^{-1}y^{-1}$ holds for all $x,y\in G$, we also know that $(xy)^{-1}=y^{-1}x^{-1}$, so $y^{-1}x^{-1}=x^{-1}y^{-1}$ and $G$ has to be Abelian.

Is this proof complete or not? Thank you.