Description: Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | necon4d.1 | |- ( ph -> ( A =/= B -> C =/= D ) ) |
|
| Assertion | necon4d | |- ( ph -> ( C = D -> A = B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | necon4d.1 | |- ( ph -> ( A =/= B -> C =/= D ) ) |
|
| 2 | 1 | necon2bd | |- ( ph -> ( C = D -> -. A =/= B ) ) |
| 3 | nne | |- ( -. A =/= B <-> A = B ) |
|
| 4 | 2 3 | syl6ib | |- ( ph -> ( C = D -> A = B ) ) |