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Theorem cadbi123i 1395
Description: Equality theorem for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
hadbii.1  |-  ( ph  <->  ps )
hadbii.2  |-  ( ch  <->  th )
hadbii.3  |-  ( ta  <->  et )
Assertion
Ref Expression
cadbi123i  |-  (cadd (
ph ,  ch ,  ta )  <-> cadd ( ps ,  th ,  et ) )

Proof of Theorem cadbi123i
StepHypRef Expression
1 hadbii.1 . . . 4  |-  ( ph  <->  ps )
21a1i 11 . . 3  |-  (  T. 
->  ( ph  <->  ps )
)
3 hadbii.2 . . . 4  |-  ( ch  <->  th )
43a1i 11 . . 3  |-  (  T. 
->  ( ch  <->  th )
)
5 hadbii.3 . . . 4  |-  ( ta  <->  et )
65a1i 11 . . 3  |-  (  T. 
->  ( ta  <->  et )
)
72, 4, 6cadbi123d 1393 . 2  |-  (  T. 
->  (cadd ( ph ,  ch ,  ta )  <-> cadd ( ps ,  th ,  et ) ) )
87trud 1333 1  |-  (cadd (
ph ,  ch ,  ta )  <-> cadd ( ps ,  th ,  et ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    T. wtru 1326  caddwcad 1389
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 179  df-or 361  df-an 362  df-xor 1315  df-tru 1329  df-cad 1391
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