Theorem frege9 44566

Description: Closed form of syl 18 with swapped antecedents. This proposition differs from frege5 44554 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 44554	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 44563	. 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)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 44544  ax-frege2 44545  ax-frege8 44563

This theorem is used by:  frege11  44568  frege10  44574  frege19  44578  frege21  44581  frege37  44594  frege56aid  44624  frege56a  44625  frege61a  44633  frege56b  44652  frege61b  44660  frege56c  44673  frege61c  44678  frege117  44734  frege130  44747  frege132  44749