Theorem frege9 44508

Description: Closed form of syl 18 with swapped antecedents. This proposition differs from frege5 44496 only in an unessential way. Identical to imim1 84. 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 44496	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 44505	. 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 44486  ax-frege2 44487  ax-frege8 44505

This theorem is referenced by:  frege11  44510  frege10  44516  frege19  44520  frege21  44523  frege37  44536  frege56aid  44566  frege56a  44567  frege61a  44575  frege56b  44594  frege61b  44602  frege56c  44615  frege61c  44620  frege117  44676  frege130  44689  frege132  44691