Are $\phi(X^2)$ and $\psi(Y^2)$ always independent where $\phi$ and $\psi$ are Borel-Measurable functions (X,Y are any two RVs)?

Question

In An Introduction to Probability and Statistics by Vijay K. Rohatgi and Ehsanes Saleh 2e, Page no 121, it is shown that if X, Y be two RVs with joint PDF, $f(x,y)$= $\frac{1+xy}{4} $, $|x|<1$ and $|y|<1 $, then X and Y are not independent but the RVs $X^2$ and $Y^2$ are independent, the concluding remark says:

Note that $\phi(X^2)$ and $\psi(Y^2)$ are independent where $\phi$ and $\psi$ are Borel-Measurable functions (X and Y are RVs)

Is it a general statement or just a comment valid for the previous question because though it seems like an important result yet it is not mentioned as a theorem. If it is indeed general, how to proceed for a proof?