Theorem negdid

Description: Distribution of negative over addition. (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	negdid	$$⊢\phi \to -\left(A+B\right)=-A+\left(-B\right)$$

Proof

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