Theorem frege9 40156

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 40144 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 40144	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 40153	. 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 40134  ax-frege2 40135  ax-frege8 40153

This theorem is referenced by:  frege11  40158  frege10  40164  frege19  40168  frege21  40171  frege37  40184  frege56aid  40214  frege56a  40215  frege61a  40223  frege56b  40242  frege61b  40250  frege56c  40263  frege61c  40268  frege117  40324  frege130  40337  frege132  40339