I have a question regarding the minimax theorem (https://en.wikipedia.org/wiki/Minimax_theorem). The minimax theorem tells us that for $x\in X, y\in Y$ where $X, Y$ are compact and closed sets, if $f(\cdot, y)$ is convex to all $y$ for fixed $x$ and $f(x, \cdot)$ is concave to all $x$ for fixed $y$, then
$\max_x\min_y f(x,y) = \min_y \max_x f(x,y)$ holds (and there exists Nash equilibrium point).
Now, my question is as follows: what happens when the two player have different objective functions, but they are still convex and concave to each variable? For instance, player X maximize $f_1(x,\cdot)$ that is concave to $x\in X$ and player Y minimizes $f_2(\cdot,y)$ that is convex to $y\in Y$. Can we still apply the minimax theorem in this case and conclude there exists a Nash equilibrium?
Regardless of whether this works or not, I want to know mathematical reasons of why this works or not, and understand. Thank you in advance :)
Update: thanks copper for your comment. In my case two $f$s are related to other: $f_2 = |f_1|$.