Theorem frege23

Description: Syllogism followed by rotation of three antecedents. Proposition 23 of Frege1879 p. 42. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege23	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜏 → 𝜑 ) → ( 𝜓 → ( 𝜒 → ( 𝜏 → 𝜃 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege18	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜏 → 𝜑 ) → ( 𝜓 → ( 𝜏 → ( 𝜒 → 𝜃 ) ) ) ) )
2		frege22	⊢ ( ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜏 → 𝜑 ) → ( 𝜓 → ( 𝜏 → ( 𝜒 → 𝜃 ) ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜏 → 𝜑 ) → ( 𝜓 → ( 𝜒 → ( 𝜏 → 𝜃 ) ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜏 → 𝜑 ) → ( 𝜓 → ( 𝜒 → ( 𝜏 → 𝜃 ) ) ) ) )