Theorem frege32 45025

Description: Deduce con1 147 from con3 154. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege32	⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))

Proof of Theorem frege32

Step	Hyp	Ref	Expression
1		ax-frege31 45024	. 2 ⊢ (¬ ¬ 𝜑 → 𝜑)
2		frege7 44998	. 2 ⊢ ((¬ ¬ 𝜑 → 𝜑) → (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))))
3	1, 2	ax-mp 5	1 ⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))

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

This proof depends on axioms:  ax-mp 5  ax-frege1 44980  ax-frege2 44981  ax-frege31 45024

This theorem is used by:  frege33  45026