Theorem nppcan2d

Description: Cancellation law for 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	nppcan2d	$$⊢\phi \to A-\left(B+C\right)+C=A-B$$

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		nppcan2	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to A-\left(B+C\right)+C=A-B$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A-\left(B+C\right)+C=A-B$$