Theorem addneintr2d

Description: Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad . Consequence of addcan2d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	muld.1	|- ( ph -> A e. CC )
	addcomd.2	|- ( ph -> B e. CC )
	addcand.3	|- ( ph -> C e. CC )
	addneintr2d.4	|- ( ph -> A =/= B )
Assertion	addneintr2d	|- ( ph -> ( A + C ) =/= ( B + C ) )

Proof

Step	Hyp	Ref	Expression
1		muld.1	|- ( ph -> A e. CC )
2		addcomd.2	|- ( ph -> B e. CC )
3		addcand.3	|- ( ph -> C e. CC )
4		addneintr2d.4	|- ( ph -> A =/= B )
5	1 2 3	addcan2d	|- ( 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 ) )