Theorem frege9 44257

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 44245 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 44245	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 44254	. 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 44235  ax-frege2 44236  ax-frege8 44254

This theorem is referenced by:  frege11  44259  frege10  44265  frege19  44269  frege21  44272  frege37  44285  frege56aid  44315  frege56a  44316  frege61a  44324  frege56b  44343  frege61b  44351  frege56c  44364  frege61c  44369  frege117  44425  frege130  44438  frege132  44440