Metamath Proof Explorer


Theorem frege46

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 ) )

Proof

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 ) )