If we assume both distributivity and the opposite of the law of signs (ie, that $-1\times-1 = -1$) for the relative integers, then we can derive that two different numbers are actually equal.
$$-2(5+-3) = -2\times2 = -4$$ but, $$-2(5+-3) = -2\times5 + -2\times-3 = -10 + -6 = -16$$
The axioms that are conventionally assumed for the integers are simply the ring axioms. My question is, if the set of "axioms" described above turns out to be inconsistent, how can we be so sure that the ring axioms aren't inconsistent as well?
I'm aware of this question. On the one hand, I'm asking if you really need to venture that deeply into proof theory just for this (seemingly simple) special case. But if you do, could you provide an example of how to apply that technique to prove this special case?