Question regarding GCD elimination

Question

I have the following expression $GCD[d b(x-y)+a(d-x y), a(x-y)+b(d-x y)]$ where x and y are variables and a and b are co-prime. Now, if I eliminate the variable $x$ from the first expression of the GCD i.e. $d b(x-y)+a(d-x y)$ , I have $GCD[(y^2-d)*(a^2-d b^2), a(x-y)+b(d-x y)]$. The question is then can I not eliminate the variable $y$ from the second expression of the GCD i.e. $a(x-y)+b(d-x y)$ so that I am left with $GCD[(y^2-d)*(a^2-d b^2), (x^2-d)*(a^2-d b^2)]$? What is wrong in doing so?