Theorem subcanad

Description: Cancellation law for subtraction. Deduction form of subcan . Generalization of subcand . (Contributed by David Moews, 28-Feb-2017)

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