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Theorem hadbi 1377
Description: The half adder is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
hadbi  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  <->  ps )  <->  ch ) )

Proof of Theorem hadbi
StepHypRef Expression
1 df-xor 1296 . 2  |-  ( ( ( ph \/_ ps ) \/_ ch )  <->  -.  (
( ph \/_ ps )  <->  ch ) )
2 df-had 1370 . 2  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph \/_ ps ) \/_ ch ) )
3 xnor 1297 . . . 4  |-  ( (
ph 
<->  ps )  <->  -.  ( ph \/_ ps ) )
43bibi1i 305 . . 3  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( -.  ( ph \/_ ps )  <->  ch )
)
5 nbbn 347 . . 3  |-  ( ( -.  ( ph \/_ ps )  <->  ch )  <->  -.  (
( ph \/_ ps )  <->  ch ) )
64, 5bitri 240 . 2  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  -.  ( ( ph \/_ ps )  <->  ch )
)
71, 2, 63bitr4i 268 1  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  <->  ps )  <->  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   \/_wxo 1295  haddwhad 1368
This theorem is referenced by:  had1  1392
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 177  df-xor 1296  df-had 1370
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