Theorem frege11

Description: Elimination of a nested antecedent as a partial converse of ja . If the proposition that ps takes place or ph does not is a sufficient condition for ch , then ps by itself is a sufficient condition for ch . Identical to jarr . Proposition 11 of Frege1879 p. 36. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege11	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege1	⊢ ( 𝜓 → ( 𝜑 → 𝜓 ) )
2		frege9	⊢ ( ( 𝜓 → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
3	1 2	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) )