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Theorem hadnot 1501
Description: The adder sum distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
Assertion
Ref Expression
hadnot  |-  ( -. hadd
( ph ,  ps ,  ch )  <-> hadd ( -.  ph ,  -.  ps ,  -.  ch ) )

Proof of Theorem hadnot
StepHypRef Expression
1 notbi 297 . . 3  |-  ( (
ph 
<->  ps )  <->  ( -.  ph  <->  -. 
ps ) )
21bibi1i 316 . 2  |-  ( ( ( ph  <->  ps )  <->  -. 
ch )  <->  ( ( -.  ph  <->  -.  ps )  <->  -. 
ch ) )
3 xor3 359 . . 3  |-  ( -.  ( ( ph  <->  ps )  <->  ch )  <->  ( ( ph  <->  ps )  <->  -.  ch )
)
4 hadbi 1497 . . 3  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  <->  ps )  <->  ch ) )
53, 4xchnxbir 311 . 2  |-  ( -. hadd
( ph ,  ps ,  ch )  <->  ( ( ph  <->  ps )  <->  -.  ch )
)
6 hadbi 1497 . 2  |-  (hadd ( -.  ph ,  -.  ps ,  -.  ch )  <->  ( ( -.  ph  <->  -.  ps )  <->  -. 
ch ) )
72, 5, 63bitr4i 281 1  |-  ( -. hadd
( ph ,  ps ,  ch )  <-> hadd ( -.  ph ,  -.  ps ,  -.  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188  haddwhad 1492
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 189  df-xor 1402  df-had 1493
This theorem is referenced by:  had0  1503
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