Theorem frege9 43824

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 43812 only in an unessential way. Identical to imim1 83. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem frege9

Step	Hyp	Ref	Expression
1		frege5 43812	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 43821	. 2 ⊢ (((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 43802  ax-frege2 43803  ax-frege8 43821

This theorem is referenced by:  frege11  43826  frege10  43832  frege19  43836  frege21  43839  frege37  43852  frege56aid  43882  frege56a  43883  frege61a  43891  frege56b  43910  frege61b  43918  frege56c  43931  frege61c  43936  frege117  43992  frege130  44005  frege132  44007