Theorem frege47 41459

Description: Deduce consequence follows from either path implied by a disjunction. If 𝜑, as well as 𝜓 is sufficient condition for 𝜒 and 𝜓 or 𝜑 takes place, then the proposition 𝜒 holds. Proposition 47 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege47	⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒)))

Proof of Theorem frege47

Step	Hyp	Ref	Expression
1		frege46 41458	. 2 ⊢ ((¬ 𝜑 → 𝜒) → ((𝜑 → 𝜒) → 𝜒))
2		frege21 41435	. 2 ⊢ (((¬ 𝜑 → 𝜒) → ((𝜑 → 𝜒) → 𝜒)) → ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒))))
3	1, 2	ax-mp 5	1 ⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 41398  ax-frege2 41399  ax-frege8 41417  ax-frege28 41438  ax-frege31 41442  ax-frege41 41453

This theorem is referenced by:  frege48  41460  frege49  41461