Theorem pncan

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

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

Proof

Step	Hyp	Ref	Expression
1		simpr	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to B\in ℂ$$
2		simpl	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A\in ℂ$$
3	1 2	addcomd	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to B+A=A+B$$
4		addcl	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A+B\in ℂ$$
5		subadd	$$⊢\left(A+B\in ℂ\wedge B\in ℂ\wedge A\in ℂ\right)\to \left(A+B-B=A\leftrightarrow B+A=A+B\right)$$
6	4 1 2 5	syl3anc	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(A+B-B=A\leftrightarrow B+A=A+B\right)$$
7	3 6	mpbird	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A+B-B=A$$