Theorem subneintrd

Description: Introducing subtraction on both sides of a statement of inequality. Contrapositive 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 )
	subneintrd.4	|- ( ph -> B =/= C )
Assertion	subneintrd	|- ( ph -> ( A - B ) =/= ( A - 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		subneintrd.4	|- ( ph -> B =/= C )
5	1 2 3	subcanad	|- ( ph -> ( ( A - B ) = ( A - C ) <-> B = C ) )
6	5	necon3bid	|- ( ph -> ( ( A - B ) =/= ( A - C ) <-> B =/= C ) )
7	4 6	mpbird	|- ( ph -> ( A - B ) =/= ( A - C ) )