Theorem frege9 44157

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 44145 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 44145	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 44154	. 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 44135  ax-frege2 44136  ax-frege8 44154

This theorem is referenced by:  frege11  44159  frege10  44165  frege19  44169  frege21  44172  frege37  44185  frege56aid  44215  frege56a  44216  frege61a  44224  frege56b  44243  frege61b  44251  frege56c  44264  frege61c  44269  frege117  44325  frege130  44338  frege132  44340