Description: If ps holds when ph occurs as well as when ph does not occur, then ps holds. If ps or ph occurs and if the occurrences of ph has ps as a necessary consequence, then ps takes place. Identical to pm2.6 . Proposition 46 of Frege1879 p. 48. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | frege46 | |- ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frege33 | |- ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) |
|
| 2 | frege45 | |- ( ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) -> ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) ) |
|
| 3 | 1 2 | ax-mp | |- ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) |