Theorem frege51 44931

Description: Compare with jaod 873. Proposition 51 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege51	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜒) → (𝜑 → ((¬ 𝜓 → 𝜃) → 𝜒))))

Proof of Theorem frege51

Step	Hyp	Ref	Expression
1		frege50 44930	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜃 → 𝜒) → ((¬ 𝜓 → 𝜃) → 𝜒)))
2		frege18 44894	. 2 ⊢ (((𝜓 → 𝜒) → ((𝜃 → 𝜒) → ((¬ 𝜓 → 𝜃) → 𝜒))) → ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜒) → (𝜑 → ((¬ 𝜓 → 𝜃) → 𝜒)))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜒) → (𝜑 → ((¬ 𝜓 → 𝜃) → 𝜒))))

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

This proof depends on axioms:  ax-mp 5  ax-frege1 44866  ax-frege2 44867  ax-frege8 44885  ax-frege28 44906  ax-frege31 44910  ax-frege41 44921

This theorem is used by:  frege128  45067