Theorem frege40 44843

Description: Anything implies pm2.18 129. 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 44842	. 2 ⊢ ((¬ 𝜓 → 𝜓) → (¬ 𝜓 → 𝜑))
2		frege35 44838	. 2 ⊢ (((¬ 𝜓 → 𝜓) → (¬ 𝜓 → 𝜑)) → (¬ 𝜑 → ((¬ 𝜓 → 𝜓) → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (¬ 𝜑 → ((¬ 𝜓 → 𝜓) → 𝜓))

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

This proof depends on axioms:  ax-mp 5  ax-frege1 44790  ax-frege2 44791  ax-frege8 44809  ax-frege28 44830  ax-frege31 44834

This theorem is used by:  frege43  44847