Why can I replace equal expressions in any mathematical statement? Is there a proof for that? Who "says" that replacing the statements is valid?
Example:
If: $$ A=B \qquad C+A^2 = D $$
Then: $$ C+B^2=D $$
Note: that $A,B,C,D$ are mathematical expressions (This is just one example, I ask why it works for every mathematical statement, not just in equations.)
Roughly speaking, when you know that two mathematical expressions are equal you know that they are different names for the same object. Then substituting one for the other is clearly OK.
See Question about properties of "equality" (which I would post as a duplicate if that question had an accepted answer).