Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | necon1bd.1 | |- ( ph -> ( A =/= B -> ps ) ) |
|
| Assertion | necon1bd | |- ( ph -> ( -. ps -> A = B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | necon1bd.1 | |- ( ph -> ( A =/= B -> ps ) ) |
|
| 2 | df-ne | |- ( A =/= B <-> -. A = B ) |
|
| 3 | 2 1 | syl5bir | |- ( ph -> ( -. A = B -> ps ) ) |
| 4 | 3 | con1d | |- ( ph -> ( -. ps -> A = B ) ) |