Metamath Proof Explorer

Theorem frege20

Description: A closed form of syl8 . Proposition 20 of Frege1879 p. 40. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Assertion frege20 
|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( th -> ta ) -> ( ph -> ( ps -> ( ch -> ta ) ) ) ) )

Proof

Step Hyp Ref Expression
1 frege19 
 |-  ( ( ps -> ( ch -> th ) ) -> ( ( th -> ta ) -> ( ps -> ( ch -> ta ) ) ) )
2 frege18 
 |-  ( ( ( ps -> ( ch -> th ) ) -> ( ( th -> ta ) -> ( ps -> ( ch -> ta ) ) ) ) -> ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( th -> ta ) -> ( ph -> ( ps -> ( ch -> ta ) ) ) ) ) )
3 1 2 ax-mp 
 |-  ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( th -> ta ) -> ( ph -> ( ps -> ( ch -> ta ) ) ) ) )