Theorem df3an2

Description: Express triple-and in terms of implication and negation. Statement in Frege1879 p. 12. (Contributed by RP, 25-Jul-2020)

	Ref	Expression
Assertion	df3an2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ¬ ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-3an	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
2		df-an	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ¬ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝜒 ) )
3		impexp	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) ) )
4	2 3	xchbinx	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ¬ ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) ) )
5	1 4	bitri	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ¬ ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) ) )