Theorem frege48 44694

Description: Closed form of syllogism with internal disjunction. If 𝜑 is a sufficient condition for the occurrence of 𝜒 or 𝜓 and if 𝜒, as well as 𝜓, is a sufficient condition for 𝜃, then 𝜑 is a sufficient condition for 𝜃. See application in frege101 44806. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege48	⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → ((𝜒 → 𝜃) → ((𝜓 → 𝜃) → (𝜑 → 𝜃))))

Proof of Theorem frege48

Step	Hyp	Ref	Expression
1		frege47 44693	. 2 ⊢ ((¬ 𝜓 → 𝜒) → ((𝜒 → 𝜃) → ((𝜓 → 𝜃) → 𝜃)))
2		frege23 44667	. 2 ⊢ (((¬ 𝜓 → 𝜒) → ((𝜒 → 𝜃) → ((𝜓 → 𝜃) → 𝜃))) → ((𝜑 → (¬ 𝜓 → 𝜒)) → ((𝜒 → 𝜃) → ((𝜓 → 𝜃) → (𝜑 → 𝜃)))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → ((𝜒 → 𝜃) → ((𝜓 → 𝜃) → (𝜑 → 𝜃))))

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:  frege101  44806