Theorem frege9 41282

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 41270 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 41270	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 41279	. 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 41260  ax-frege2 41261  ax-frege8 41279

This theorem is referenced by:  frege11  41284  frege10  41290  frege19  41294  frege21  41297  frege37  41310  frege56aid  41340  frege56a  41341  frege61a  41349  frege56b  41368  frege61b  41376  frege56c  41389  frege61c  41394  frege117  41450  frege130  41463  frege132  41465