I am trying to understand a proof where it is used that a point $P=[y_0:y_1] \in \mathbb{P}_1(\mathbb{Q}),\,y_1,y_2 \in \mathbb{Q}$ can be written in integral coprime form, so $x0,x1\in\mathbb{Z},\, gcd(x0,x1) = 1$ and $[y_0,y_1] = [x_0,x_1]$.
I tried to start with $y0 = \frac{a}{b}$ such that $ gcd(a,b) = 1$ and $y1 = \frac{c}{d}$ such that $gcd(c,d)=1$. Then we have $[\frac{a}{b},\frac{c}{d}] = [ad,bc]$ where now $ad,bc \in \mathbb{Z}$. If I can now show that $gcd(ad,bc)=1$ I would be done but this is where I got stuck.