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Theorem hadnot 1436
Description: The half adder distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
hadnot  |-  ( -. hadd
( ph ,  ps ,  ch )  <-> hadd ( -.  ph ,  -.  ps ,  -.  ch ) )

Proof of Theorem hadnot
StepHypRef Expression
1 xorneg 1363 . . . 4  |-  ( ( -.  ph  \/_  -.  ps ) 
<->  ( ph  \/_  ps ) )
2 biid 236 . . . 4  |-  ( -. 
ch 
<->  -.  ch )
31, 2xorbi12i 1364 . . 3  |-  ( ( ( -.  ph  \/_  -.  ps )  \/_  -.  ch ) 
<->  ( ( ph  \/_  ps )  \/_  -.  ch )
)
4 xorneg2 1362 . . 3  |-  ( ( ( ph  \/_  ps )  \/_  -.  ch )  <->  -.  ( ( ph  \/_  ps )  \/_  ch ) )
53, 4bitr2i 250 . 2  |-  ( -.  ( ( ph  \/_  ps )  \/_  ch )  <->  ( ( -.  ph  \/_  -.  ps )  \/_  -.  ch ) )
6 df-had 1422 . . 3  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  \/_ 
ps )  \/_  ch ) )
76notbii 296 . 2  |-  ( -. hadd
( ph ,  ps ,  ch )  <->  -.  ( ( ph  \/_  ps )  \/_  ch ) )
8 df-had 1422 . 2  |-  (hadd ( -.  ph ,  -.  ps ,  -.  ch )  <->  ( ( -.  ph  \/_  -.  ps )  \/_  -.  ch ) )
95, 7, 83bitr4i 277 1  |-  ( -. hadd
( ph ,  ps ,  ch )  <-> hadd ( -.  ph ,  -.  ps ,  -.  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/_ wxo 1351  haddwhad 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-xor 1352  df-had 1422
This theorem is referenced by:  had0  1446
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