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Theorem 3anor 981
Description: Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.)
Assertion
Ref Expression
3anor  |-  ( (
ph  /\  ps  /\  ch ) 
<->  -.  ( -.  ph  \/  -.  ps  \/  -.  ch ) )

Proof of Theorem 3anor
StepHypRef Expression
1 df-3an 967 . 2  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
2 anor 489 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ch )  <->  -.  ( -.  ( ph  /\  ps )  \/  -.  ch )
)
3 ianor 488 . . . . 5  |-  ( -.  ( ph  /\  ps ) 
<->  ( -.  ph  \/  -.  ps ) )
43orbi1i 520 . . . 4  |-  ( ( -.  ( ph  /\  ps )  \/  -.  ch )  <->  ( ( -. 
ph  \/  -.  ps )  \/  -.  ch ) )
52, 4xchbinx 310 . . 3  |-  ( ( ( ph  /\  ps )  /\  ch )  <->  -.  (
( -.  ph  \/  -.  ps )  \/  -.  ch ) )
6 df-3or 966 . . 3  |-  ( ( -.  ph  \/  -.  ps  \/  -.  ch )  <->  ( ( -.  ph  \/  -.  ps )  \/  -.  ch ) )
75, 6xchbinxr 311 . 2  |-  ( ( ( ph  /\  ps )  /\  ch )  <->  -.  ( -.  ph  \/  -.  ps  \/  -.  ch ) )
81, 7bitri 249 1  |-  ( (
ph  /\  ps  /\  ch ) 
<->  -.  ( -.  ph  \/  -.  ps  \/  -.  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965
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 185  df-or 370  df-an 371  df-3or 966  df-3an 967
This theorem is referenced by:  3ianor  982  ne3anior  2717
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