Theorem frege9 43774

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 43762 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 43762	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 43771	. 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 43752  ax-frege2 43753  ax-frege8 43771

This theorem is referenced by:  frege11  43776  frege10  43782  frege19  43786  frege21  43789  frege37  43802  frege56aid  43832  frege56a  43833  frege61a  43841  frege56b  43860  frege61b  43868  frege56c  43881  frege61c  43886  frege117  43942  frege130  43955  frege132  43957