Theorem pnpcand

Description: Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	pncand.2	$$⊢\phi \to B\in ℂ$$
	subaddd.3	$$⊢\phi \to C\in ℂ$$
Assertion	pnpcand	$$⊢\phi \to A+B-\left(A+C\right)=B-C$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		subaddd.3	$$⊢\phi \to C\in ℂ$$
4		pnpcan	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to A+B-\left(A+C\right)=B-C$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A+B-\left(A+C\right)=B-C$$