Metamath Proof Explorer


Theorem frege37

Description: If ch is a necessary consequence of the occurrence of ps or ph , then ch is a necessary consequence of ph alone. Similar to a closed form of orcs . Proposition 37 of Frege1879 p. 46. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Assertion frege37
|- ( ( ( -. ph -> ps ) -> ch ) -> ( ph -> ch ) )

Proof

Step Hyp Ref Expression
1 frege36
 |-  ( ph -> ( -. ph -> ps ) )
2 frege9
 |-  ( ( ph -> ( -. ph -> ps ) ) -> ( ( ( -. ph -> ps ) -> ch ) -> ( ph -> ch ) ) )
3 1 2 ax-mp
 |-  ( ( ( -. ph -> ps ) -> ch ) -> ( ph -> ch ) )