Theorem frege9

Description: Closed form of syl with swapped antecedents. This proposition differs from frege5 only in an unessential way. Identical to imim1 . Proposition 9 of Frege1879 p. 35. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege9	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege5	⊢ ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )
2		ax-frege8	⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )