Theorem frege45 38143

Description: Deduce pm2.6 182 from con1 143. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege45	⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))

Proof of Theorem frege45

Step	Hyp	Ref	Expression
1		frege44 38142	. 2 ⊢ ((¬ 𝜓 → 𝜑) → ((𝜑 → 𝜓) → 𝜓))
2		frege5 38094	. 2 ⊢ (((¬ 𝜓 → 𝜑) → ((𝜑 → 𝜓) → 𝜓)) → (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege28 38124  ax-frege31 38128  ax-frege41 38139

This theorem is referenced by:  frege46  38144