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Theorem frege37 37954
Description: If 𝜒 is a necessary consequence of the occurrence of 𝜓 or 𝜑, then 𝜒 is a necessary consequence of 𝜑 alone. Similar to a closed form of orcs 409. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege37 (((¬ 𝜑𝜓) → 𝜒) → (𝜑𝜒))

Proof of Theorem frege37
StepHypRef Expression
1 frege36 37953 . 2 (𝜑 → (¬ 𝜑𝜓))
2 frege9 37926 . 2 ((𝜑 → (¬ 𝜑𝜓)) → (((¬ 𝜑𝜓) → 𝜒) → (𝜑𝜒)))
31, 2ax-mp 5 1 (((¬ 𝜑𝜓) → 𝜒) → (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-frege1 37904  ax-frege2 37905  ax-frege8 37923  ax-frege28 37944  ax-frege31 37948
This theorem is referenced by:  frege106  38083
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