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 ) ) |
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 ) ) |