Theorem subcan2d

Description: Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016)

	Ref	Expression
Hypotheses	negidd.1	|- ( ph -> A e. CC )
	pncand.2	|- ( ph -> B e. CC )
	subaddd.3	|- ( ph -> C e. CC )
	subcan2d.4	|- ( ph -> ( A - C ) = ( B - C ) )
Assertion	subcan2d	|- ( ph -> A = B )

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		subcan2d.4	|- ( ph -> ( A - C ) = ( B - C ) )
5		subcan2	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) = ( B - C ) <-> A = B ) )
6	1 2 3 5	syl3anc	|- ( ph -> ( ( A - C ) = ( B - C ) <-> A = B ) )
7	4 6	mpbid	|- ( ph -> A = B )