Metamath Proof Explorer


Theorem frege9

Description: Closed form of syl with swapped antecedents. This proposition differs from frege5 only in an unessential way. Identical to imim1 . Proposition 9 of Frege1879 p. 35. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Assertion frege9 ( ( 𝜑𝜓 ) → ( ( 𝜓𝜒 ) → ( 𝜑𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 frege5 ( ( 𝜓𝜒 ) → ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) )
2 ax-frege8 ( ( ( 𝜓𝜒 ) → ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) ) → ( ( 𝜑𝜓 ) → ( ( 𝜓𝜒 ) → ( 𝜑𝜒 ) ) ) )
3 1 2 ax-mp ( ( 𝜑𝜓 ) → ( ( 𝜓𝜒 ) → ( 𝜑𝜒 ) ) )