Prove that $G$ is an Abelian group

Question

I would need help for this one. I've been stuck on this exercise:

If $G$ is a group with $a,b,c,x,y \in G$ where $xay=bac \implies xy = bc$. Prove that this group $G$ is Abelian.

Can someone give me a tip on where to start? I literally have no clue how to start proving this. Thanks and sorry for bad formatting. I'm new.

Answer 1

Let $a=x^{-1}$, $b=y$ and $c=x$. The remaining statement is quantified over only $x$ and $y$. See what it says.