Theorem subnegd

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	subnegd	$$⊢\phi \to A-\left(-B\right)=A+B$$

Proof

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