Theorem addsub4i

Description: Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999)

	Ref	Expression
Hypotheses	negidi.1	$$⊢A\in ℂ$$
	pncan3i.2	$$⊢B\in ℂ$$
	subadd.3	$$⊢C\in ℂ$$
	addsub4i.4	$$⊢D\in ℂ$$
Assertion	addsub4i	$$⊢A+B-\left(C+D\right)=\left(A-C\right)+B-D$$

Proof

Step	Hyp	Ref	Expression
1		negidi.1	$$⊢A\in ℂ$$
2		pncan3i.2	$$⊢B\in ℂ$$
3		subadd.3	$$⊢C\in ℂ$$
4		addsub4i.4	$$⊢D\in ℂ$$
5		addsub4	$$⊢\left(\left(A\in ℂ\wedge B\in ℂ\right)\wedge \left(C\in ℂ\wedge D\in ℂ\right)\right)\to A+B-\left(C+D\right)=\left(A-C\right)+B-D$$
6	1 2 3 4 5	mp4an	$$⊢A+B-\left(C+D\right)=\left(A-C\right)+B-D$$