integration by substitution from two variables to one

In lecture notes I have seen the substitution $t = x\cdot y$ applied for the following integral: $$\int_{0}^{\infty}e^{-\frac{y^2}{2}}\int_{0}^{\infty}e^{-\frac{(xy)^2}{2}}\cdot y \: dxdy = \left( \int_{0}^{\infty}e^{-\frac{t^2}{2}}dt \right) ^2$$ Why can two variables be substituted by only one and what are the intermediate steps?

You make a change of variable $(t,y) = (xy,y)$. The Jacobian matrix is $$\begin{pmatrix} \frac{1}{y} & -\frac{t}{y^2}\\ 0 & 1\\ \end{pmatrix} $$ so its determinant is $\frac{1}{y}$.

And the double integral becomes $$\iint_{0<t,y<+\infty} \exp{\left(-\frac{y^2}{2}-\frac{t^2}{2} \right)}y\frac{1}{y}dtdy =\left( \int_{0}^{\infty}e^{-\frac{t^2}{2}}dt \right) ^2$$