We have $a|bc$ so $bc=na$ with $n$ an integer. Dividing both sides by $a$ we have $$\frac{bc}{a}=n$$ But $\gcd(a,b)=1$ implies $b/a$ is only an integer if $a=1$ So $$\frac{bc}{a} = b \frac{c}{a} = n $$ which means $a$ must divide $c$

Note that this proof relies on the Fundamental Theorem of Arithmetic. The alternative proof of Euclid's lemma avoids this.