Theorem pnpcand

Description: Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		negidd.1	|- ( ph -> A e. CC )
2		pncand.2	|- ( ph -> B e. CC )
3		subaddd.3	|- ( ph -> C e. CC )
4		pnpcan	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A + C ) ) = ( B - C ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( ( A + B ) - ( A + C ) ) = ( B - C ) )