Theorem addcan2d

Description: Cancellation law for addition. Theorem I.1 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	muld.1	|- ( ph -> A e. CC )
	addcomd.2	|- ( ph -> B e. CC )
	addcand.3	|- ( ph -> C e. CC )
Assertion	addcan2d	|- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )

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		addcan2	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) = ( B + C ) <-> A = B ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )