Description: If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d . Contrapositive of subeq0bd . (Contributed by David Moews, 28-Feb-2017)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | negidd.1 | |- ( ph -> A e. CC ) |
|
| pncand.2 | |- ( ph -> B e. CC ) |
||
| subne0ad.3 | |- ( ph -> ( A - B ) =/= 0 ) |
||
| Assertion | subne0ad | |- ( ph -> A =/= B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | negidd.1 | |- ( ph -> A e. CC ) |
|
| 2 | pncand.2 | |- ( ph -> B e. CC ) |
|
| 3 | subne0ad.3 | |- ( ph -> ( A - B ) =/= 0 ) |
|
| 4 | 1 2 | subeq0ad | |- ( ph -> ( ( A - B ) = 0 <-> A = B ) ) |
| 5 | 4 | necon3bid | |- ( ph -> ( ( A - B ) =/= 0 <-> A =/= B ) ) |
| 6 | 3 5 | mpbid | |- ( ph -> A =/= B ) |