I was trying to prove that $-(x + y) = -x - y$ and as you can see in the image below, I took the liberty of using the $-$ symbol as a number and applying the associative law with it. Is it kosher in all rigorousness given the axioms professional mathematicians use?
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$$(x+y)+(-x-y)=(y+x)+(-x-y)=y+(x-x)+(-y)=y+0+(-y)=0$$ So $(x+y)$ is the additive inverse of $(-x-y)$. Hence $-(x+y)=-x-y$
Well the LHS of your equation is just saying "the additive inverse of $x+y$".
So all you have to show is that the additive inverse of $x+y$ really is the RHS of the equation, i.e. $-x-y$, then by uniqueness of inverses in a group the two must be equal.