Theorem addcanad

Description: Cancelling a term on the left-hand side of a sum in an equality. 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 )
	addcanad.4	|- ( ph -> ( A + B ) = ( A + C ) )
Assertion	addcanad	|- ( ph -> 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		addcanad.4	|- ( ph -> ( A + B ) = ( A + C ) )
5	1 2 3	addcand	|- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) )
6	4 5	mpbid	|- ( ph -> B = C )