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Theorem cadbi123d 1373
Description: Equality theorem for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
hadbid.1  |-  ( ph  ->  ( ps  <->  ch )
)
hadbid.2  |-  ( ph  ->  ( th  <->  ta )
)
hadbid.3  |-  ( ph  ->  ( et  <->  ze )
)
Assertion
Ref Expression
cadbi123d  |-  ( ph  ->  (cadd ( ps ,  th ,  et )  <-> cadd ( ch ,  ta ,  ze ) ) )

Proof of Theorem cadbi123d
StepHypRef Expression
1 hadbid.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
2 hadbid.2 . . . 4  |-  ( ph  ->  ( th  <->  ta )
)
31, 2anbi12d 691 . . 3  |-  ( ph  ->  ( ( ps  /\  th )  <->  ( ch  /\  ta ) ) )
4 hadbid.3 . . . 4  |-  ( ph  ->  ( et  <->  ze )
)
51, 2xorbi12d 1306 . . . 4  |-  ( ph  ->  ( ( ps  \/_  th )  <->  ( ch  \/_  ta ) ) )
64, 5anbi12d 691 . . 3  |-  ( ph  ->  ( ( et  /\  ( ps  \/_  th )
)  <->  ( ze  /\  ( ch  \/_  ta )
) ) )
73, 6orbi12d 690 . 2  |-  ( ph  ->  ( ( ( ps 
/\  th )  \/  ( et  /\  ( ps  \/_  th ) ) )  <->  ( ( ch  /\  ta )  \/  ( ze  /\  ( ch  \/_  ta ) ) ) ) )
8 df-cad 1371 . 2  |-  (cadd ( ps ,  th ,  et )  <->  ( ( ps 
/\  th )  \/  ( et  /\  ( ps  \/_  th ) ) ) )
9 df-cad 1371 . 2  |-  (cadd ( ch ,  ta ,  ze )  <->  ( ( ch 
/\  ta )  \/  ( ze  /\  ( ch  \/_  ta ) ) ) )
107, 8, 93bitr4g 279 1  |-  ( ph  ->  (cadd ( ps ,  th ,  et )  <-> cadd ( ch ,  ta ,  ze ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/_ wxo 1295  caddwcad 1369
This theorem is referenced by:  cadbi123i  1375  sadfval  12659  sadcp1  12662
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-or 359  df-an 360  df-xor 1296  df-cad 1371
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