I think saying that a theorem is true presumes the axioms from which it was proven and so the entire system is "true everywhere forever".

I often find it helpful to think of chess as my axiomatic system. When we say the king is in checkmate, it presumes that we accept all the underlying rules of chess. And these pieces that theoretically form a checkmate will always do so forever... Assuming the usual rules of chess, assuming they're unchanging, etc.

When you put things in terms of chess, these "deep" statements about "math" often become banal. And it works for any game that's a "formal system" (eg. most board games).