Theorem npcan

Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	npcan	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A-B+B=A$$

Proof

Step	Hyp	Ref	Expression
1		subcl	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A-B\in ℂ$$
2		simpr	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to B\in ℂ$$
3	1 2	addcomd	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A-B+B=B+A-B$$
4		pncan3	$$⊢\left(B\in ℂ\wedge A\in ℂ\right)\to B+A-B=A$$
5	4	ancoms	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to B+A-B=A$$
6	3 5	eqtrd	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A-B+B=A$$