Metamath Proof Explorer

Theorem frege45

Description: Deduce pm2.6 from con1 . Proposition 45 of Frege1879 p. 47. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Assertion frege45 
|- ( ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) -> ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) )

Proof

Step Hyp Ref Expression
1 frege44 
 |-  ( ( -. ps -> ph ) -> ( ( ph -> ps ) -> ps ) )
2 frege5 
 |-  ( ( ( -. ps -> ph ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) -> ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) ) )
3 1 2 ax-mp 
 |-  ( ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) -> ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) )