Theorem frege47

Description: Deduce consequence follows from either path implied by a disjunction. If ph , as well as ps is sufficient condition for ch and ps or ph takes place, then the proposition ch holds. Proposition 47 of Frege1879 p. 48. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege47	⊢ ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜒 ) → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege46	⊢ ( ( ¬ 𝜑 → 𝜒 ) → ( ( 𝜑 → 𝜒 ) → 𝜒 ) )
2		frege21	⊢ ( ( ( ¬ 𝜑 → 𝜒 ) → ( ( 𝜑 → 𝜒 ) → 𝜒 ) ) → ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜒 ) → 𝜒 ) ) ) )
3	1 2	ax-mp	⊢ ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜒 ) → 𝜒 ) ) )