Proposition: For all integers m, n, p, q (m-n) - (p-q) = (m+q) - (n+p)

Question

I have to prove the proposition above using field axioms for integers and the definition of subtraction, which is m - n is defined to be m+(-n)

So far I only have

(m-n)-(p-q) = (m+(-n)) - (p+(-q)) by the definition of subtraction

I don't know where to go from here; I feel like I can get the end result by using associative axiom for addition and commutative axiom for addition, but I don't know where to begin.

Now I have

(m-n) - (p-q) = (m+(-n)) - (p+(-q)) by definition of subtraction

= (m+(-n)) + (-(p+(-q)) by definition of subtraction

= (m+(-n)) + (-p + - (-q)) by previous propositions

= (m+(-n)) + (-p + q) by previous propositions

Now I don't know where to go because I can't use commutative axiom on subtraction

Answer 1

Basically all you need to show is –(p-q)=(-p)+q. (1)

Since 0 = (p-q) + (-(p-q)) (additive inverse), we get –p = -q +(-(p-q)). Now substitute for (–p) in (1).

Then m+q-n-p=(m+q)-(n+p) follows. (since -n-p+(n+p) = 0 implies -n-p = -(n+p) (additive inverse))