Metamath Proof Explorer


Theorem frege50

Description: Closed form of jaoi . Proposition 50 of Frege1879 p. 49. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Assertion frege50 ( ( 𝜑𝜓 ) → ( ( 𝜒𝜓 ) → ( ( ¬ 𝜑𝜒 ) → 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 frege49 ( ( ¬ 𝜑𝜒 ) → ( ( 𝜑𝜓 ) → ( ( 𝜒𝜓 ) → 𝜓 ) ) )
2 frege17 ( ( ( ¬ 𝜑𝜒 ) → ( ( 𝜑𝜓 ) → ( ( 𝜒𝜓 ) → 𝜓 ) ) ) → ( ( 𝜑𝜓 ) → ( ( 𝜒𝜓 ) → ( ( ¬ 𝜑𝜒 ) → 𝜓 ) ) ) )
3 1 2 ax-mp ( ( 𝜑𝜓 ) → ( ( 𝜒𝜓 ) → ( ( ¬ 𝜑𝜒 ) → 𝜓 ) ) )