Theorem subcand

Description: Cancellation law for 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 )
	subcand.4	|- ( ph -> ( A - B ) = ( A - C ) )
Assertion	subcand	|- ( ph -> 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		subcand.4	|- ( ph -> ( A - B ) = ( A - C ) )
5		subcan	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = ( A - C ) <-> B = C ) )
6	1 2 3 5	syl3anc	|- ( ph -> ( ( A - B ) = ( A - C ) <-> B = C ) )
7	4 6	mpbid	|- ( ph -> B = C )