Theorem frege19

Description: A closed form of syl6 . Proposition 19 of Frege1879 p. 39. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege19	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜒 → 𝜃 ) → ( 𝜑 → ( 𝜓 → 𝜃 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege9	⊢ ( ( 𝜓 → 𝜒 ) → ( ( 𝜒 → 𝜃 ) → ( 𝜓 → 𝜃 ) ) )
2		frege18	⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜒 → 𝜃 ) → ( 𝜓 → 𝜃 ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜒 → 𝜃 ) → ( 𝜑 → ( 𝜓 → 𝜃 ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜒 → 𝜃 ) → ( 𝜑 → ( 𝜓 → 𝜃 ) ) ) )