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Theorem 3ioran 952
Description: Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
Assertion
Ref Expression
3ioran  |-  ( -.  ( ph  \/  ps  \/  ch )  <->  ( -.  ph 
/\  -.  ps  /\  -.  ch ) )

Proof of Theorem 3ioran
StepHypRef Expression
1 ioran 477 . . 3  |-  ( -.  ( ph  \/  ps ) 
<->  ( -.  ph  /\  -.  ps ) )
21anbi1i 677 . 2  |-  ( ( -.  ( ph  \/  ps )  /\  -.  ch ) 
<->  ( ( -.  ph  /\ 
-.  ps )  /\  -.  ch ) )
3 ioran 477 . . 3  |-  ( -.  ( ( ph  \/  ps )  \/  ch ) 
<->  ( -.  ( ph  \/  ps )  /\  -.  ch ) )
4 df-3or 937 . . 3  |-  ( (
ph  \/  ps  \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
53, 4xchnxbir 301 . 2  |-  ( -.  ( ph  \/  ps  \/  ch )  <->  ( -.  ( ph  \/  ps )  /\  -.  ch ) )
6 df-3an 938 . 2  |-  ( ( -.  ph  /\  -.  ps  /\ 
-.  ch )  <->  ( ( -.  ph  /\  -.  ps )  /\  -.  ch )
)
72, 5, 63bitr4i 269 1  |-  ( -.  ( ph  \/  ps  \/  ch )  <->  ( -.  ph 
/\  -.  ps  /\  -.  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    /\ w3a 936
This theorem is referenced by:  3oran  953  cadnot  1403  fbunfip  17889
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 178  df-or 360  df-an 361  df-3or 937  df-3an 938
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