Question about properties of "equality"

Question

There is one thing that is used in a lot proofs but I have never heard of it as an axiom or a theorem. I'm talking about the fact that if $x$ and $y$ are numbers such that $$x=y$$ then $$f(x)=f(y).$$ And it seems true but I have never heard of it as an axiom (or theorem but I don't think that it can be a theorem). The only way (which I see) to prove it, is to somehow use the Peano axioms and one of ways to construct natural numbers but I don't imagine it in practice. So is there such an axiom or theorem?

Answer 1

I think the answer to your question lies in the meaning of "equality". The statement $$ x = y $$ says (in everyday mathematics) that "$x$" and "$y$" are different names for the same object. So of course it follows that $$ f(x) = f(y) \ . $$

This doesn't depend on $x$ and $y$ being numbers, or even on $f$ being a function. It can be any expression in $x$ and $y$.

I say "in everyday mathematics" because if you're studying foundations you need to be much more careful about things and the names of things.

For an extended discussion, see this nice essay by Barry Mazur: http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf