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Theorem cadnot 1436
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 983 . . 3  |-  ( -.  ( ( ph  /\  ps )  \/  ( ph  /\  ch )  \/  ( ps  /\  ch ) )  <->  ( -.  ( ph  /\  ps )  /\  -.  ( ph  /\  ch )  /\  -.  ( ps  /\  ch ) ) )
2 ianor 488 . . . 4  |-  ( -.  ( ph  /\  ps ) 
<->  ( -.  ph  \/  -.  ps ) )
3 ianor 488 . . . 4  |-  ( -.  ( ph  /\  ch ) 
<->  ( -.  ph  \/  -.  ch ) )
4 ianor 488 . . . 4  |-  ( -.  ( ps  /\  ch ) 
<->  ( -.  ps  \/  -.  ch ) )
52, 3, 43anbi123i 1176 . . 3  |-  ( ( -.  ( ph  /\  ps )  /\  -.  ( ph  /\  ch )  /\  -.  ( ps  /\  ch ) )  <->  ( ( -.  ph  \/  -.  ps )  /\  ( -.  ph  \/  -.  ch )  /\  ( -.  ps  \/  -.  ch ) ) )
61, 5bitri 249 . 2  |-  ( -.  ( ( ph  /\  ps )  \/  ( ph  /\  ch )  \/  ( ps  /\  ch ) )  <->  ( ( -.  ph  \/  -.  ps )  /\  ( -.  ph  \/  -.  ch )  /\  ( -.  ps  \/  -.  ch ) ) )
7 cador 1432 . . 3  |-  (cadd (
ph ,  ps ,  ch )  <->  ( ( ph  /\ 
ps )  \/  ( ph  /\  ch )  \/  ( ps  /\  ch ) ) )
87notbii 296 . 2  |-  ( -. cadd
( ph ,  ps ,  ch )  <->  -.  ( ( ph  /\  ps )  \/  ( ph  /\  ch )  \/  ( ps  /\ 
ch ) ) )
9 cadan 1433 . 2  |-  (cadd ( -.  ph ,  -.  ps ,  -.  ch )  <->  ( ( -.  ph  \/  -.  ps )  /\  ( -.  ph  \/  -.  ch )  /\  ( -.  ps  \/  -.  ch ) ) )
106, 8, 93bitr4i 277 1  |-  ( -. cadd
( ph ,  ps ,  ch )  <-> cadd ( -.  ph ,  -.  ps ,  -.  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965  caddwcad 1420
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  df-xor 1351  df-cad 1422
This theorem is referenced by: (None)
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