Theorem df3or2

Description: Express triple-or in terms of implication and negation. Statement in Frege1879 p. 11. (Contributed by RP, 25-Jul-2020)

	Ref	Expression
Assertion	df3or2	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ¬ 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-3or	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) )
2		df-or	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) ↔ ( ¬ ( 𝜑 ∨ 𝜓 ) → 𝜒 ) )
3		ioran	⊢ ( ¬ ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 ∧ ¬ 𝜓 ) )
4	3	imbi1i	⊢ ( ( ¬ ( 𝜑 ∨ 𝜓 ) → 𝜒 ) ↔ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) )
5		impexp	⊢ ( ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) ↔ ( ¬ 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) )
6	4 5	bitri	⊢ ( ( ¬ ( 𝜑 ∨ 𝜓 ) → 𝜒 ) ↔ ( ¬ 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) )
7	2 6	bitri	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) ↔ ( ¬ 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) )
8	1 7	bitri	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ¬ 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) )