Metamath Proof Explorer


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 ) )