This is a follow-up question from Proof predicate in PA and stronger system
Suppose that two theories $T_1$ and $T_2$ share the same language - thus only axioms differ such that $T_2$ is a stronger theory relative to $T_1$.
Do/can $T_1$ and $T_2$ share the same proof predicate $Proof(x,y)$ where $x$ is Godel encoding of proof of Godel decoding of $y$? If not, why would it be? From my understanding, $Proof$ predicate is really about decoding $x$ and $y$, demonstrating that $x$ follows the right rule of inference or deduction and arriving at Godel decoding of $y$. And adding axioms does not necessarily change rule of inference or deduction. Thus, it seems that proof predicate does not have to change.