Theorem df3an2 41377

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

Assertion

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

Proof of Theorem df3an2

Step	Hyp	Ref	Expression
1		df-3an 1088	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
2		df-an 397	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ ((𝜑 ∧ 𝜓) → ¬ 𝜒))
3		impexp 451	. . 3 ⊢ (((𝜑 ∧ 𝜓) → ¬ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
4	2, 3	xchbinx 334	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))
5	1, 4	bitri 274	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088

This theorem is referenced by: (None)