How to prove that $ F[x,x^{-1},y,z]/(z^2-xy)\cong F[x,x^{-1}]\otimes_F\left(F[y,z]\big{/}(z^2-y)\right)$?

Question

Let $I=(z^2-xy)\subseteq F[x,x^{-1},y,z]$ for some algebraically closed field $F$, and $J=(z^2-y)\subseteq F[y,z]$. Then how to prove that $$ F[x,x^{-1},y,z]\big{/}I\cong F[x,x^{-1}]\otimes_F\left(F[y,z]\big{/}J\right)? $$ Here is what I tried: define the map $\varphi:F[x,x^{-1},y,z]\to F[x,x^{-1}]\otimes_F\left(F[y,z]\big{/}J\right), x\mapsto x\otimes 1, y\mapsto1\otimes(y+J), z\mapsto 1\otimes(z+J)$. Then I tried to prove that $\varphi(I)=0$, but somehow that is not as easy as I thought.