Theorem frege37 44839

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 889. 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 44838	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))
2		frege9 44811	. 2 ⊢ ((𝜑 → (¬ 𝜑 → 𝜓)) → (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜒)))
3	1, 2	ax-mp 5	1 ⊢ (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 44789  ax-frege2 44790  ax-frege8 44808  ax-frege28 44829  ax-frege31 44833

This theorem is used by:  frege106  44968