Theorem frege30 40198

Description: Commuted, closed form of con3d 155. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege30	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))

Proof of Theorem frege30

Step	Hyp	Ref	Expression
1		frege29 40197	. 2 ⊢ ((𝜓 → (𝜑 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))
2		frege10 40186	. 2 ⊢ (((𝜓 → (𝜑 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑))) → ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 40156  ax-frege2 40157  ax-frege8 40175  ax-frege28 40196

This theorem is referenced by:  frege59a  40243  frege59b  40270  frege59c  40288