Theorem frege37

Description: If ch is a necessary consequence of the occurrence of ps or ph , then ch is a necessary consequence of ph alone. Similar to a closed form of orcs . Proposition 37 of Frege1879 p. 46. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege37	⊢ ( ( ( ¬ 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		frege36	⊢ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) )
2		frege9	⊢ ( ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) → ( ( ( ¬ 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
3	1 2	ax-mp	⊢ ( ( ( ¬ 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) )