Theorem frege9 44400

Description: Closed form of syl 18 with swapped antecedents. This proposition differs from frege5 44388 only in an unessential way. Identical to imim1 84. 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 44388	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 44397	. 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 44378  ax-frege2 44379  ax-frege8 44397

This theorem is referenced by:  frege11  44402  frege10  44408  frege19  44412  frege21  44415  frege37  44428  frege56aid  44458  frege56a  44459  frege61a  44467  frege56b  44486  frege61b  44494  frege56c  44507  frege61c  44512  frege117  44568  frege130  44581  frege132  44583