Theorem frege9 43825

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 43813 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 43813	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 43822	. 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 43803  ax-frege2 43804  ax-frege8 43822

This theorem is referenced by:  frege11  43827  frege10  43833  frege19  43837  frege21  43840  frege37  43853  frege56aid  43883  frege56a  43884  frege61a  43892  frege56b  43911  frege61b  43919  frege56c  43932  frege61c  43937  frege117  43993  frege130  44006  frege132  44008