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Theorem cadnot 1404
Description: The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
cadnot  |-  ( -. cadd
( ph ,  ps ,  ch )  <-> cadd ( -.  ph ,  -.  ps ,  -.  ch ) )

Proof of Theorem cadnot
StepHypRef Expression
1 3ioran 953 . . 3  |-  ( -.  ( ( ph  /\  ps )  \/  ( ph  /\  ch )  \/  ( ps  /\  ch ) )  <->  ( -.  ( ph  /\  ps )  /\  -.  ( ph  /\  ch )  /\  -.  ( ps  /\  ch ) ) )
2 ianor 476 . . . 4  |-  ( -.  ( ph  /\  ps ) 
<->  ( -.  ph  \/  -.  ps ) )
3 ianor 476 . . . 4  |-  ( -.  ( ph  /\  ch ) 
<->  ( -.  ph  \/  -.  ch ) )
4 ianor 476 . . . 4  |-  ( -.  ( ps  /\  ch ) 
<->  ( -.  ps  \/  -.  ch ) )
52, 3, 43anbi123i 1143 . . 3  |-  ( ( -.  ( ph  /\  ps )  /\  -.  ( ph  /\  ch )  /\  -.  ( ps  /\  ch ) )  <->  ( ( -.  ph  \/  -.  ps )  /\  ( -.  ph  \/  -.  ch )  /\  ( -.  ps  \/  -.  ch ) ) )
61, 5bitri 242 . 2  |-  ( -.  ( ( ph  /\  ps )  \/  ( ph  /\  ch )  \/  ( ps  /\  ch ) )  <->  ( ( -.  ph  \/  -.  ps )  /\  ( -.  ph  \/  -.  ch )  /\  ( -.  ps  \/  -.  ch ) ) )
7 cador 1401 . . 3  |-  (cadd (
ph ,  ps ,  ch )  <->  ( ( ph  /\ 
ps )  \/  ( ph  /\  ch )  \/  ( ps  /\  ch ) ) )
87notbii 289 . 2  |-  ( -. cadd
( ph ,  ps ,  ch )  <->  -.  ( ( ph  /\  ps )  \/  ( ph  /\  ch )  \/  ( ps  /\ 
ch ) ) )
9 cadan 1402 . 2  |-  (cadd ( -.  ph ,  -.  ps ,  -.  ch )  <->  ( ( -.  ph  \/  -.  ps )  /\  ( -.  ph  \/  -.  ch )  /\  ( -.  ps  \/  -.  ch ) ) )
106, 8, 93bitr4i 270 1  |-  ( -. cadd
( ph ,  ps ,  ch )  <-> cadd ( -.  ph ,  -.  ps ,  -.  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 936    /\ w3a 937  caddwcad 1389
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 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-xor 1315  df-cad 1391
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