elementary number theory - how to start a proof of a system of congruences

Let $a$, $b$, $m$, and $n$ be integers with $m > 0$, $n >0$, and $gcd(m,n) = 1$. Then the system $x\equiv a$ (mod n) and $x\equiv b$ (mod m) has a unique solution modulo mn.

This is not the Chinese Remainder Theorem just yet. That is the next proof. This is a proof leading up to it.

Help please!

Answer

Hint $\ $ If $\,x',x\,$ are two solutions then $\,m,n\mid x'-x\,$ so $\ \ldots\mid x'-x$