Theorem frege47 45041

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 45040	. 2 ⊢ ((¬ 𝜑 → 𝜒) → ((𝜑 → 𝜒) → 𝜒))
2		frege21 45017	. 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)))

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

This proof depends on axioms:  ax-mp 5  ax-frege1 44980  ax-frege2 44981  ax-frege8 44999  ax-frege28 45020  ax-frege31 45024  ax-frege41 45035

This theorem is used by:  frege48  45042  frege49  45043