Theorem frege49

Description: Closed form of deduction with disjunction. Proposition 49 of Frege1879 p. 49. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege49	⊢ ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege47	⊢ ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜒 ) → 𝜒 ) ) )
2		frege12	⊢ ( ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜒 ) → 𝜒 ) ) ) → ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) ) )
3	1 2	ax-mp	⊢ ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )