Theorem frege48

Description: Closed form of syllogism with internal disjunction. If ph is a sufficient condition for the occurrence of ch or ps and if ch , as well as ps , is a sufficient condition for th , then ph is a sufficient condition for th . See application in frege101 . Proposition 48 of Frege1879 p. 49. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege48	⊢ ( ( 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) → ( ( 𝜒 → 𝜃 ) → ( ( 𝜓 → 𝜃 ) → ( 𝜑 → 𝜃 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege47	⊢ ( ( ¬ 𝜓 → 𝜒 ) → ( ( 𝜒 → 𝜃 ) → ( ( 𝜓 → 𝜃 ) → 𝜃 ) ) )
2		frege23	⊢ ( ( ( ¬ 𝜓 → 𝜒 ) → ( ( 𝜒 → 𝜃 ) → ( ( 𝜓 → 𝜃 ) → 𝜃 ) ) ) → ( ( 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) → ( ( 𝜒 → 𝜃 ) → ( ( 𝜓 → 𝜃 ) → ( 𝜑 → 𝜃 ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) → ( ( 𝜒 → 𝜃 ) → ( ( 𝜓 → 𝜃 ) → ( 𝜑 → 𝜃 ) ) ) )