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Theorem frege48 40082
 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 40194. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege48 ((𝜑 → (¬ 𝜓𝜒)) → ((𝜒𝜃) → ((𝜓𝜃) → (𝜑𝜃))))

Proof of Theorem frege48
StepHypRef Expression
1 frege47 40081 . 2 ((¬ 𝜓𝜒) → ((𝜒𝜃) → ((𝜓𝜃) → 𝜃)))
2 frege23 40055 . 2 (((¬ 𝜓𝜒) → ((𝜒𝜃) → ((𝜓𝜃) → 𝜃))) → ((𝜑 → (¬ 𝜓𝜒)) → ((𝜒𝜃) → ((𝜓𝜃) → (𝜑𝜃)))))
31, 2ax-mp 5 1 ((𝜑 → (¬ 𝜓𝜒)) → ((𝜒𝜃) → ((𝜓𝜃) → (𝜑𝜃))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4 This theorem was proved from axioms:  ax-mp 5  ax-frege1 40020  ax-frege2 40021  ax-frege8 40039  ax-frege28 40060  ax-frege31 40064  ax-frege41 40075 This theorem is referenced by:  frege101  40194
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