Prove: $\frac{(2px)!}{((px)!)^2}\equiv\frac{(2x)!}{((x)!)^2}\pmod{p^2}$

Question

How can I prove the following, where $p$ is a prime and $x$ a positive integer?

$$\dfrac{(2px)!}{((px)!)^2}\equiv\dfrac{(2x)!}{((x)!)^2}\pmod{p^2}$$

I'm not sure if it is actually true, but I tested for small numbers and it checked.