Theorem frege48 37166

Description: Closed form of syllogism with internal disjunction. If 𝜑 is a sufficient condition for the occurence 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 37278. 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 37165	. 2 ⊢ ((¬ 𝜓 → 𝜒) → ((𝜒 → 𝜃) → ((𝜓 → 𝜃) → 𝜃)))
2		frege23 37139	. 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))))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege28 37144  ax-frege31 37148  ax-frege41 37159

This theorem is referenced by:  frege101  37278