Theorem frege9 43910

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 43898 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 43898	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 43907	. 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 43888  ax-frege2 43889  ax-frege8 43907

This theorem is referenced by:  frege11  43912  frege10  43918  frege19  43922  frege21  43925  frege37  43938  frege56aid  43968  frege56a  43969  frege61a  43977  frege56b  43996  frege61b  44004  frege56c  44017  frege61c  44022  frege117  44078  frege130  44091  frege132  44093