Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | necon4bd.1 | |- ( ph -> ( -. ps -> A =/= B ) ) |
|
| Assertion | necon4bd | |- ( ph -> ( A = B -> ps ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | necon4bd.1 | |- ( ph -> ( -. ps -> A =/= B ) ) |
|
| 2 | 1 | necon2bd | |- ( ph -> ( A = B -> -. -. ps ) ) |
| 3 | notnotr | |- ( -. -. ps -> ps ) |
|
| 4 | 2 3 | syl6 | |- ( ph -> ( A = B -> ps ) ) |