Theorem frege49 40206

Description: Closed form of deduction with disjunction. Proposition 49 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege49	⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))

Proof of Theorem frege49

Step	Hyp	Ref	Expression
1		frege47 40204	. 2 ⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒)))
2		frege12 40166	. 2 ⊢ (((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒))) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜒) → ((𝜓 → 𝜒) → 𝜒))))
3	1, 2	ax-mp 5	1 ⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 40143  ax-frege2 40144  ax-frege8 40162  ax-frege28 40183  ax-frege31 40187  ax-frege41 40198

This theorem is referenced by:  frege50  40207