Theorem subneintr2d

Description: Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d . (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 )
	subneintr2d.4	|- ( ph -> A =/= B )
Assertion	subneintr2d	|- ( ph -> ( 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		subneintr2d.4	|- ( ph -> A =/= B )
5	1 2 3	subcan2ad	|- ( ph -> ( ( A - C ) = ( B - C ) <-> A = B ) )
6	5	necon3bid	|- ( ph -> ( ( A - C ) =/= ( B - C ) <-> A =/= B ) )
7	4 6	mpbird	|- ( ph -> ( A - C ) =/= ( B - C ) )