How to formally show that the choice of a coordinate system in a physical system can be arbitrary?

My question is:

How to rigorously (and elegantly) show that one can choose any arbitrary coordinate system in a physical system and arrive at the same conclusion about the system no matter what one chooses?

I saw a proof in Feynman's book "SIX NOT-SO- EASY PIECES" but that showed the equivalence of rotated coordinate axes. Also, I was not convinced that was rigorous.