Theorem pnpncand

Description: Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017)

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

Proof

Step	Hyp	Ref	Expression
1		pnpncand.1	$$⊢\phi \to A\in ℂ$$
2		pnpncand.2	$$⊢\phi \to B\in ℂ$$
3		pnpncand.3	$$⊢\phi \to C\in ℂ$$
4	2 3	subcld	$$⊢\phi \to B-C\in ℂ$$
5	1 4	addcld	$$⊢\phi \to A+B-C\in ℂ$$
6	5 2 3	subsub2d	$$⊢\phi \to A+\left(B-C\right)-\left(B-C\right)=A+\left(B-C\right)+\left(C-B\right)$$
7	1 4	pncand	$$⊢\phi \to A+\left(B-C\right)-\left(B-C\right)=A$$
8	6 7	eqtr3d	$$⊢\phi \to A+\left(B-C\right)+\left(C-B\right)=A$$