Theorem frege9 43794

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 43782 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 43782	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 43791	. 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 43772  ax-frege2 43773  ax-frege8 43791

This theorem is referenced by:  frege11  43796  frege10  43802  frege19  43806  frege21  43809  frege37  43822  frege56aid  43852  frege56a  43853  frege61a  43861  frege56b  43880  frege61b  43888  frege56c  43901  frege61c  43906  frege117  43962  frege130  43975  frege132  43977