Theorem neg2subd

Description: Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	pncand.2	$$⊢\phi \to B\in ℂ$$
Assertion	neg2subd	$$⊢\phi \to -A-\left(-B\right)=B-A$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		neg2sub	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to -A-\left(-B\right)=B-A$$
4	1 2 3	syl2anc	$$⊢\phi \to -A-\left(-B\right)=B-A$$