Theorem pnpncand

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

	Ref	Expression
Hypotheses	pnpncand.1	|- ( ph -> A e. CC )
	pnpncand.2	|- ( ph -> B e. CC )
	pnpncand.3	|- ( ph -> C e. CC )
Assertion	pnpncand	|- ( ph -> ( ( A + ( B - C ) ) + ( C - B ) ) = A )

Proof

Step	Hyp	Ref	Expression
1		pnpncand.1	|- ( ph -> A e. CC )
2		pnpncand.2	|- ( ph -> B e. CC )
3		pnpncand.3	|- ( ph -> C e. CC )
4	2 3	subcld	|- ( ph -> ( B - C ) e. CC )
5	1 4	addcld	|- ( ph -> ( A + ( B - C ) ) e. CC )
6	5 2 3	subsub2d	|- ( ph -> ( ( A + ( B - C ) ) - ( B - C ) ) = ( ( A + ( B - C ) ) + ( C - B ) ) )
7	1 4	pncand	|- ( ph -> ( ( A + ( B - C ) ) - ( B - C ) ) = A )
8	6 7	eqtr3d	|- ( ph -> ( ( A + ( B - C ) ) + ( C - B ) ) = A )