Metamath Proof Explorer

Theorem frege23

Description: Syllogism followed by rotation of three antecedents. Proposition 23 of Frege1879 p. 42. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Assertion frege23 
|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ta -> ph ) -> ( ps -> ( ch -> ( ta -> th ) ) ) ) )

Proof

Step Hyp Ref Expression
1 frege18 
 |-  ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ta -> ph ) -> ( ps -> ( ta -> ( ch -> th ) ) ) ) )
2 frege22 
 |-  ( ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ta -> ph ) -> ( ps -> ( ta -> ( ch -> th ) ) ) ) ) -> ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ta -> ph ) -> ( ps -> ( ch -> ( ta -> th ) ) ) ) ) )
3 1 2 ax-mp 
 |-  ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ta -> ph ) -> ( ps -> ( ch -> ( ta -> th ) ) ) ) )