Theorem frege9 40513

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 40501 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 40501	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 40510	. 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 40491  ax-frege2 40492  ax-frege8 40510

This theorem is referenced by:  frege11  40515  frege10  40521  frege19  40525  frege21  40528  frege37  40541  frege56aid  40571  frege56a  40572  frege61a  40580  frege56b  40599  frege61b  40607  frege56c  40620  frege61c  40625  frege117  40681  frege130  40694  frege132  40696