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Theorem 3ioran 962
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 726 . . 3  |-  ( -.  ( ph  \/  ps ) 
<->  ( -.  ph  /\  -.  ps ) )
21anbi1i 453 . 2  |-  ( ( -.  ( ph  \/  ps )  /\  -.  ch ) 
<->  ( ( -.  ph  /\ 
-.  ps )  /\  -.  ch ) )
3 ioran 726 . . 3  |-  ( -.  ( ( ph  \/  ps )  \/  ch ) 
<->  ( -.  ( ph  \/  ps )  /\  -.  ch ) )
4 df-3or 948 . . 3  |-  ( (
ph  \/  ps  \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
53, 4xchnxbir 655 . 2  |-  ( -.  ( ph  \/  ps  \/  ch )  <->  ( -.  ( ph  \/  ps )  /\  -.  ch ) )
6 df-3an 949 . 2  |-  ( ( -.  ph  /\  -.  ps  /\ 
-.  ch )  <->  ( ( -.  ph  /\  -.  ps )  /\  -.  ch )
)
72, 5, 63bitr4i 211 1  |-  ( -.  ( ph  \/  ps  \/  ch )  <->  ( -.  ph 
/\  -.  ps  /\  -.  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    \/ wo 682    \/ w3o 946    /\ w3a 947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949
This theorem is referenced by:  ne3anior  2373
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