I read the proof of this theorem in Apostol's Mathematical Analysis. It was not $100 \%$ clear so I re-did it in a way that makes more sense to me. Please tell me if this is an acceptable proof.
$a|bc \implies bc=ak$ for some integer $k$
$\gcd(a,b)=1 \implies 1 = ax + by \implies c = axc + byc$ for some integers $x$ and $y$.
Multiplying both sides of the first equation by $y$ we get $$bcy = aky$$
Substituting this into the second equation we get $$c = axc + (aky)$$
Factoring out $a$ we get $c = a(xc+ky)$ which implies that $a|c$.