Why can I replace equal expressions in mathematical statements?

Question

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.)

What definition of equality are you using? In some formal systems, being able to substitute is exactly the definition of equality, so you get this ability tautologically. — Trebor, Dec 11 '23 at 15:11
Substitution for equality: "if $A=B$, then $(\varphi(A) \to \varphi(B))$ where $\varphi$ is a formula (a mathematical statement). — Mauro ALLEGRANZA, Dec 11 '23 at 15:15
This is a great question, but it is a duplicate. Note that the duplicate I've linked to mentions fields specifically in the question, but the answer is really at this level of generality. — Noah Schweber, Dec 11 '23 at 16:54

score 1 · Answer 1 · answered Dec 11 '23 at 15:12

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).

Why can I replace equal expressions in mathematical statements?

1 Answers1