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Theorem 3ioran 989
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 488 . . 3  |-  ( -.  ( ph  \/  ps ) 
<->  ( -.  ph  /\  -.  ps ) )
21anbi1i 693 . 2  |-  ( ( -.  ( ph  \/  ps )  /\  -.  ch ) 
<->  ( ( -.  ph  /\ 
-.  ps )  /\  -.  ch ) )
3 ioran 488 . . 3  |-  ( -.  ( ( ph  \/  ps )  \/  ch ) 
<->  ( -.  ( ph  \/  ps )  /\  -.  ch ) )
4 df-3or 972 . . 3  |-  ( (
ph  \/  ps  \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
53, 4xchnxbir 307 . 2  |-  ( -.  ( ph  \/  ps  \/  ch )  <->  ( -.  ( ph  \/  ps )  /\  -.  ch ) )
6 df-3an 973 . 2  |-  ( ( -.  ph  /\  -.  ps  /\ 
-.  ch )  <->  ( ( -.  ph  /\  -.  ps )  /\  -.  ch )
)
72, 5, 63bitr4i 277 1  |-  ( -.  ( ph  \/  ps  \/  ch )  <->  ( -.  ph 
/\  -.  ps  /\  -.  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 366    /\ wa 367    \/ w3o 970    /\ w3a 971
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 368  df-an 369  df-3or 972  df-3an 973
This theorem is referenced by:  3oran  990  cadnot  1471  cadnotOLD  1472  fbunfip  20478  wwlknndef  24883  wwlknfi  24884  clwwlknndef  24919  frgraregord013  25264  wl-nfeqfb  30195
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