In Tao's book Analysis 1, he writes:
Thus, from the point of view of logic, we can define equality on a [remark by myself: I think he forgot the word "type of object" here] however we please, so long as it obeys the reflexive, symmetry, and transitive axioms, and it is consistent with all other operations on the class of objects under discussion in the sense that the substitution axiom was true for all of those operations.
Does he mean that, if one wants to define define equality on a specific type of object (like functions, ordered pairs, for example), one has to check that these axioms of equality (he refers to these four axioms of equality as "symmetry", "reflexivity", "transitivity", and "substitution") hold in the sense that one has to prove them? It seems so, because of these two passages:
[In section 3.3 Functions] We observe that functions obey the axiom of substitution: if $x=x'$, then $f(x) = f(x')$ (why?).
(My answer would be "because that's an axiom", but Tao apparently wouldn't accept that.)
And after defining equality of sets ($A=B:\iff \forall x(x\in A\iff x\in B)$), Tao writes (on page 39):
One can easily verify that this notion of equality is reflexive, symmetric, and transitive (Exercise 3.1.1). Observe that if $x\in A$ and $A = B$, then $x\in B$, by Definition 3.1.4. Thus the "is an element of" relation $\in $ obeys the axiom of substitution
So he gives the exercise to prove the axioms of equality for sets. Why does one has to prove axioms? Or, put differently: if one can prove these things, why does he state them as axioms?