Metamath Proof Explorer


Theorem frege37

Description: If ch is a necessary consequence of the occurrence of ps or ph , then ch is a necessary consequence of ph alone. Similar to a closed form of orcs . Proposition 37 of Frege1879 p. 46. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Assertion frege37 ( ( ( ¬ 𝜑𝜓 ) → 𝜒 ) → ( 𝜑𝜒 ) )

Proof

Step Hyp Ref Expression
1 frege36 ( 𝜑 → ( ¬ 𝜑𝜓 ) )
2 frege9 ( ( 𝜑 → ( ¬ 𝜑𝜓 ) ) → ( ( ( ¬ 𝜑𝜓 ) → 𝜒 ) → ( 𝜑𝜒 ) ) )
3 1 2 ax-mp ( ( ( ¬ 𝜑𝜓 ) → 𝜒 ) → ( 𝜑𝜒 ) )