Theorem addneintrd

Description: Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad . Consequence of addcand . (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 )
	addneintrd.4	|- ( ph -> B =/= C )
Assertion	addneintrd	|- ( ph -> ( A + B ) =/= ( A + 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		addneintrd.4	|- ( ph -> B =/= C )
5	1 2 3	addcand	|- ( 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 ) )