Prove two double integrals are equal

I need to prove:

Where , is a function of and , and is merely piecewise continuous, so it doesn't have to be differentiable at certain points, and is an arbitrary function also of and , which is infinitely differentiable and has compact support.

My Attempt:

I tried to change variables by using and , and I later realized that it basically became an exercise in trying to prove which didn't work out.

Can anyone give me a hint of some kind on how to try and proceed? Any suggestions would be greatly appreciated. Thanks in advance.