Theorem frege51 41463

Description: Compare with jaod 856. 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 41462	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜃 → 𝜒) → ((¬ 𝜓 → 𝜃) → 𝜒)))
2		frege18 41426	. 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))))

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:  frege128  41599