Prove Euler's totient corollary

Question

I saw the following statement on the Wikipedia page of Euler's theorem, and I was wondering how we could prove it. Basically, I don't see how I can make the step from one mod to the other mod. Any help to start out would be much appreciated!

Answer 1

We have $$a^x-a^y=a^y\cdot (a^{x-y}-1)\equiv a^y(1-1)=0 \mod n$$

when $\phi(n)$ divides $x-y$ because we have $a^{\phi(n)}\equiv 1\mod n$ and with $x-y=k\phi(n)$ we get $a^{x-y}=(a^{\phi(n)})^k\equiv 1^k=1\mod n$