Theorem frege9 45004

Description: Closed form of syl 18 with swapped antecedents. This proposition differs from frege5 44992 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 44992	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 45001	. 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 44982  ax-frege2 44983  ax-frege8 45001

This theorem is used by:  frege11  45006  frege10  45012  frege19  45016  frege21  45019  frege37  45032  frege56aid  45062  frege56a  45063  frege61a  45071  frege56b  45090  frege61b  45098  frege56c  45111  frege61c  45116  frege117  45172  frege130  45185  frege132  45187