Theorem frege32 40188

Description: Deduce con1 148 from con3 156. 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 40187	. 2 ⊢ (¬ ¬ 𝜑 → 𝜑)
2		frege7 40161	. 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)))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 40143  ax-frege2 40144  ax-frege31 40187

This theorem is referenced by:  frege33  40189