Theorem frege30 44969

Description: Commuted, closed form of con3d 153. 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 44968	. 2 ⊢ ((𝜓 → (𝜑 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))
2		frege10 44957	. 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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 44927  ax-frege2 44928  ax-frege8 44946  ax-frege28 44967

This theorem is used by:  frege59a  45014  frege59b  45041  frege59c  45059