I am just starting Apostol's Calculus Vol. 1, and I have no experience with rigorous mathematics. In his introduction, the first proof he gives is that if $a+b=a+c$, then $b=c$.
It says, "Given $a+b=a+c$. By Axiom 5, there is a real number $y$ such that $y+a=0$. Since sums are uniquely determined, we have $y+(a+b)=y+(a+c)$..." This part confuses me because it appears that he is essentially using substitution, but substitution hasn't been mentioned at all.
Before the theorem he is proving, he says it is assumed sums are uniquely determined and gives only these six axioms for the real-number system: commutative laws, associative laws, distributive law, existence of identity elements, and existence of reciprocals.
I'm not sure how the allowance of substitution follows from any of these. I mean, he explicitly states it follows from the assumption that sums are uniquely determined, but I don't follow how that allows substitution.
Edit: Apparently it's a property of equality called the substitution property which states "For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b) (if both sides make sense, i.e. are well-formed)." https://en.wikipedia.org/wiki/Equality_%28mathematics%29