Theorem frege9 42867

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 42855 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 42855	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 42864	. 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 42845  ax-frege2 42846  ax-frege8 42864

This theorem is referenced by:  frege11  42869  frege10  42875  frege19  42879  frege21  42882  frege37  42895  frege56aid  42925  frege56a  42926  frege61a  42934  frege56b  42953  frege61b  42961  frege56c  42974  frege61c  42979  frege117  43035  frege130  43048  frege132  43050