Theorem 3anor 948

Description: Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.)

Assertion

Ref	Expression
3anor	⊢ ((φ ∧ ψ ∧ χ) ↔ ¬ (¬ φ ∨ ¬ ψ ∨ ¬ χ))

Proof of Theorem 3anor

Step	Hyp	Ref	Expression
1		df-3an 936	. 2 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
2		anor 475	. . . 4 ⊢ (((φ ∧ ψ) ∧ χ) ↔ ¬ (¬ (φ ∧ ψ) ∨ ¬ χ))
3		ianor 474	. . . . 5 ⊢ (¬ (φ ∧ ψ) ↔ (¬ φ ∨ ¬ ψ))
4	3	orbi1i 506	. . . 4 ⊢ ((¬ (φ ∧ ψ) ∨ ¬ χ) ↔ ((¬ φ ∨ ¬ ψ) ∨ ¬ χ))
5	2, 4	xchbinx 301	. . 3 ⊢ (((φ ∧ ψ) ∧ χ) ↔ ¬ ((¬ φ ∨ ¬ ψ) ∨ ¬ χ))
6		df-3or 935	. . 3 ⊢ ((¬ φ ∨ ¬ ψ ∨ ¬ χ) ↔ ((¬ φ ∨ ¬ ψ) ∨ ¬ χ))
7	5, 6	xchbinxr 302	. 2 ⊢ (((φ ∧ ψ) ∧ χ) ↔ ¬ (¬ φ ∨ ¬ ψ ∨ ¬ χ))
8	1, 7	bitri 240	1 ⊢ ((φ ∧ ψ ∧ χ) ↔ ¬ (¬ φ ∨ ¬ ψ ∨ ¬ χ))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∨ w3o 933   ∧ w3a 934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936

This theorem is used by:  3ianor  949  ne3anior  2603