MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hadbi Structured version   Visualization version   Unicode version

Theorem hadbi 1500
Description: The adder sum 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 1405 . 2  |-  ( ( ( ph  \/_  ps )  \/_  ch )  <->  -.  (
( ph  \/_  ps )  <->  ch ) )
2 df-had 1496 . 2  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  \/_ 
ps )  \/_  ch ) )
3 xnor 1406 . . . 4  |-  ( (
ph 
<->  ps )  <->  -.  ( ph  \/_  ps ) )
43bibi1i 316 . . 3  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( -.  ( ph  \/_  ps )  <->  ch )
)
5 nbbn 360 . . 3  |-  ( ( -.  ( ph  \/_  ps ) 
<->  ch )  <->  -.  (
( ph  \/_  ps )  <->  ch ) )
64, 5bitri 253 . 2  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  -.  ( ( ph  \/_  ps )  <->  ch )
)
71, 2, 63bitr4i 281 1  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  <->  ps )  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    \/_ wxo 1404  haddwhad 1495
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 1405  df-had 1496
This theorem is referenced by:  hadnot  1504  had1  1505
  Copyright terms: Public domain W3C validator