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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
StepHypRef 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))
43orbi1i 506 . . . 4 ((¬ (φ ψ) ¬ χ) ↔ ((¬ φ ¬ ψ) ¬ χ))
52, 4xchbinx 301 . . 3 (((φ ψ) χ) ↔ ¬ ((¬ φ ¬ ψ) ¬ χ))
6 df-3or 935 . . 3 ((¬ φ ¬ ψ ¬ χ) ↔ ((¬ φ ¬ ψ) ¬ χ))
75, 6xchbinxr 302 . 2 (((φ ψ) χ) ↔ ¬ (¬ φ ¬ ψ ¬ χ))
81, 7bitri 240 1 ((φ ψ χ) ↔ ¬ (¬ φ ¬ ψ ¬ χ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∨ w3o 933   ∧ w3a 934 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  3ianor  949  ne3anior  2602
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