Theorem frege9 40966

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 40954 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 40954	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		ax-frege8 40963	. 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 40944  ax-frege2 40945  ax-frege8 40963

This theorem is referenced by:  frege11  40968  frege10  40974  frege19  40978  frege21  40981  frege37  40994  frege56aid  41024  frege56a  41025  frege61a  41033  frege56b  41052  frege61b  41060  frege56c  41073  frege61c  41078  frege117  41134  frege130  41147  frege132  41149