Theorem frege49 44695

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 44693	. 2 ⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒)))
2		frege12 44655	. 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)))

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

This proof depends on axioms:  ax-mp 5  ax-frege1 44632  ax-frege2 44633  ax-frege8 44651  ax-frege28 44672  ax-frege31 44676  ax-frege41 44687

This theorem is used by:  frege50  44696