Theorem frege20

Description: A closed form of syl8 . Proposition 20 of Frege1879 p. 40. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege20	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜃 → 𝜏 ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege19	⊢ ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) → ( ( 𝜃 → 𝜏 ) → ( 𝜓 → ( 𝜒 → 𝜏 ) ) ) )
2		frege18	⊢ ( ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) → ( ( 𝜃 → 𝜏 ) → ( 𝜓 → ( 𝜒 → 𝜏 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜃 → 𝜏 ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜃 → 𝜏 ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) ) ) )