Metamath Proof Explorer


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 ) )