Is it true that $\int_y\int_x|\delta(x=y)f(y)|^2\,dydx=\int_y|f(y)|^2\,dy$?

Question

I want to prove or disprove $$\int_y\int_x|\delta(x=y)f(y)|^2\,dydx=\int_y|f(y)|^2\,dy.$$

One "proof" is $$\int_y\int_x|\delta(x=y)f(y)|^2\,dydx=\int_y\int_x\delta(x=y)g(x)\,dxdy=\int_yg(y)\,dy=\int_y\delta(y=y)|f(y)|^2\,dy=\int_y|f(y)|^2\,dy$$where $g(x)=\delta(x=y)|f(x)|^2$.

I am unsure about the last step.