Theorem frege37 41337

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 871. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege37	⊢ (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜒))

Proof of Theorem frege37

Step	Hyp	Ref	Expression
1		frege36 41336	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))
2		frege9 41309	. 2 ⊢ ((𝜑 → (¬ 𝜑 → 𝜓)) → (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜒)))
3	1, 2	ax-mp 5	1 ⊢ (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 41287  ax-frege2 41288  ax-frege8 41306  ax-frege28 41327  ax-frege31 41331

This theorem is referenced by:  frege106  41466