Theorem frege50 43844

Description: Closed form of jaoi 857. Proposition 50 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege50	⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((¬ 𝜑 → 𝜒) → 𝜓)))

Proof of Theorem frege50

Step	Hyp	Ref	Expression
1		frege49 43843	. 2 ⊢ ((¬ 𝜑 → 𝜒) → ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → 𝜓)))
2		frege17 43811	. 2 ⊢ (((¬ 𝜑 → 𝜒) → ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((¬ 𝜑 → 𝜒) → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((¬ 𝜑 → 𝜒) → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 43780  ax-frege2 43781  ax-frege8 43799  ax-frege28 43820  ax-frege31 43824  ax-frege41 43835

This theorem is referenced by:  frege51  43845