Metamath Proof Explorer


Theorem frege16

Description: A closed form of com34 . Proposition 16 of Frege1879 p. 38. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Assertion frege16 ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜃𝜏 ) ) ) ) → ( 𝜑 → ( 𝜓 → ( 𝜃 → ( 𝜒𝜏 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 frege12 ( ( 𝜓 → ( 𝜒 → ( 𝜃𝜏 ) ) ) → ( 𝜓 → ( 𝜃 → ( 𝜒𝜏 ) ) ) )
2 frege5 ( ( ( 𝜓 → ( 𝜒 → ( 𝜃𝜏 ) ) ) → ( 𝜓 → ( 𝜃 → ( 𝜒𝜏 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜃𝜏 ) ) ) ) → ( 𝜑 → ( 𝜓 → ( 𝜃 → ( 𝜒𝜏 ) ) ) ) ) )
3 1 2 ax-mp ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜃𝜏 ) ) ) ) → ( 𝜑 → ( 𝜓 → ( 𝜃 → ( 𝜒𝜏 ) ) ) ) )