Theorem frege45 37163

Description: Deduce pm2.6 181 from con1 142. 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 37162	. 2 ⊢ ((¬ 𝜓 → 𝜑) → ((𝜑 → 𝜓) → 𝜓))
2		frege5 37114	. 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 37104  ax-frege2 37105  ax-frege8 37123  ax-frege28 37144  ax-frege31 37148  ax-frege41 37159

This theorem is referenced by:  frege46  37164