Theorem frege40 40209

Description: Anything implies pm2.18 128. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege40	⊢ (¬ 𝜑 → ((¬ 𝜓 → 𝜓) → 𝜓))

Proof of Theorem frege40

Step	Hyp	Ref	Expression
1		frege39 40208	. 2 ⊢ ((¬ 𝜓 → 𝜓) → (¬ 𝜓 → 𝜑))
2		frege35 40204	. 2 ⊢ (((¬ 𝜓 → 𝜓) → (¬ 𝜓 → 𝜑)) → (¬ 𝜑 → ((¬ 𝜓 → 𝜓) → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (¬ 𝜑 → ((¬ 𝜓 → 𝜓) → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 40156  ax-frege2 40157  ax-frege8 40175  ax-frege28 40196  ax-frege31 40200

This theorem is referenced by:  frege43  40213