This is a question I have regard the proof of the cancellation law for addition in Apostol's Calculus. We are told that the sum of two real numbers x and y is x+y and that it is uniquely determined. We are also given a few axioms. The ones that matter are the commutative law, associative law and the existence of negatives.
Cancellation law for addition: If $a + b = a + c$, then $b = c$.
We assume that $a + b = a + c$. By the Existence of Negatives Axiom, we know that there is a number $y$ such that $y + a = 0$.
My question regards this step:
Since sums are uniquely determined we have $y + (a + b) = y + (a + c)$. Why is it allowed to do this step? What is the reasoning behind it?