Theorem 3ianor 949

Description: Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Assertion

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

Proof of Theorem 3ianor

Step	Hyp	Ref	Expression
1		3anor 948	. . 3 ⊢ ((φ ∧ ψ ∧ χ) ↔ ¬ (¬ φ ∨ ¬ ψ ∨ ¬ χ))
2	1	con2bii 322	. 2 ⊢ ((¬ φ ∨ ¬ ψ ∨ ¬ χ) ↔ ¬ (φ ∧ ψ ∧ χ))
3	2	bicomi 193	1 ⊢ (¬ (φ ∧ ψ ∧ χ) ↔ (¬ φ ∨ ¬ ψ ∨ ¬ χ))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ w3o 933   ∧ w3a 934

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 177  df-or 359  df-an 360  df-3or 935  df-3an 936

This theorem is referenced by: (None)