In the following proofs I have read here and here, it appears to me, both assume what is being proven is already true and use it in the proof itself.
What I understand so far:
1) Subtraction can be defined as addition such that
$ a - b = a + (-b) $
2) From this definition
$a - (-b) = a + - (-b)$
3) Using the definition of the additive inverse, then for any number c, -c satisfies the equation
$c + (-c) = 0$
4) $x = - (-b)$ is the number that solves the equation $(-b) + x = 0$, meaning $-(-b) = b$ for it to be true.
Step 4 is where my confusion occurs. $x = - (-b)$ still, but couldn't it have been assumed that
$-(-b) = -b$ and instead the equation $b + x = 0$ would be true?
Point being that in the proof(s) it's already assumed that $-(-b) = b$, and that knowledge is used to construct an equation that is tailored to prove that fact. But if the assumption were $-(-b) = -b$, then the equation can be rewritten to prove that as well.
Can someone please clear my confusion with detailed and explicit steps? I would be very grateful as this has been occupying my thoughts for the last few days.
Nothing is being assumed beyond the definition of additive inverse. We can prove that $-(-b) = b$ -- i.e., that $b$ is an additive inverse of an additive inverse of $b$ -- from the definition of additive inverse.
The fact that $-b$ is an additive inverse of $b$ means, by definition, $$ b + (-b) = 0$$ An additive inverse of $-b$ is whatever number $x$ satisfies $$ x + (-b) = 0$$ But comparing this to the first equation, which we know to be true, we see that $x = b$ satisfies this second equation. Thus $b$ is an additive inverse of $-b$, i.e., $b = -(-b)$.
Note: Unfortunately the symbol $-$ is being used in two distinct ways, to denote an additive inverse and to denote subtraction. This is standard, so I didn't want to change it, but that may be part of what is confusing here.