Theorem frege9 40248

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 40236 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 40236	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 40245	. 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 40226  ax-frege2 40227  ax-frege8 40245

This theorem is referenced by:  frege11  40250  frege10  40256  frege19  40260  frege21  40263  frege37  40276  frege56aid  40306  frege56a  40307  frege61a  40315  frege56b  40334  frege61b  40342  frege56c  40355  frege61c  40360  frege117  40416  frege130  40429  frege132  40431