Theorem frege29 44617

Description: Closed form of con3d 153. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege29	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))

Proof of Theorem frege29

Step	Hyp	Ref	Expression
1		ax-frege28 44616	. 2 ⊢ ((𝜓 → 𝜒) → (¬ 𝜒 → ¬ 𝜓))
2		frege5 44586	. 2 ⊢ (((𝜓 → 𝜒) → (¬ 𝜒 → ¬ 𝜓)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))

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

This proof depends on axioms:  ax-mp 5  ax-frege1 44576  ax-frege2 44577  ax-frege28 44616

This theorem is used by:  frege30  44618