Theorem negdii

Description: Distribution of negative over addition. (Contributed by NM, 28-Jul-1999) (Proof shortened by OpenAI, 25-Mar-2011)

	Ref	Expression
Hypotheses	negidi.1	$$⊢A\in ℂ$$
	pncan3i.2	$$⊢B\in ℂ$$
Assertion	negdii	$$⊢-\left(A+B\right)=-A+\left(-B\right)$$

Proof

Step	Hyp	Ref	Expression
1		negidi.1	$$⊢A\in ℂ$$
2		pncan3i.2	$$⊢B\in ℂ$$
3		negdi	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to -\left(A+B\right)=-A+\left(-B\right)$$
4	1 2 3	mp2an	$$⊢-\left(A+B\right)=-A+\left(-B\right)$$