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Theorem hadbi 1454
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 1364 . 2  |-  ( ( ( ph  \/_  ps )  \/_  ch )  <->  -.  (
( ph  \/_  ps )  <->  ch ) )
2 df-had 1447 . 2  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  \/_ 
ps )  \/_  ch ) )
3 xnor 1365 . . . 4  |-  ( (
ph 
<->  ps )  <->  -.  ( ph  \/_  ps ) )
43bibi1i 314 . . 3  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( -.  ( ph  \/_  ps )  <->  ch )
)
5 nbbn 358 . . 3  |-  ( ( -.  ( ph  \/_  ps ) 
<->  ch )  <->  -.  (
( ph  \/_  ps )  <->  ch ) )
64, 5bitri 249 . 2  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  -.  ( ( ph  \/_  ps )  <->  ch )
)
71, 2, 63bitr4i 277 1  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  <->  ps )  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/_ wxo 1363  haddwhad 1445
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 1364  df-had 1447
This theorem is referenced by:  had1  1470
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