Theorem frege9 42611

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 42599 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 42599	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 42608	. 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 42589  ax-frege2 42590  ax-frege8 42608

This theorem is referenced by:  frege11  42613  frege10  42619  frege19  42623  frege21  42626  frege37  42639  frege56aid  42669  frege56a  42670  frege61a  42678  frege56b  42697  frege61b  42705  frege56c  42718  frege61c  42723  frege117  42779  frege130  42792  frege132  42794