MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cad0 Unicode version

Theorem cad0 1390
Description: If one parameter is false, the adder carry is true exactly when both of the other two parameters are true. (Contributed by Mario Carneiro, 8-Sep-2016.)
Assertion
Ref Expression
cad0  |-  ( -. 
ch  ->  (cadd ( ph ,  ps ,  ch )  <->  (
ph  /\  ps )
) )

Proof of Theorem cad0
StepHypRef Expression
1 df-cad 1371 . 2  |-  (cadd (
ph ,  ps ,  ch )  <->  ( ( ph  /\ 
ps )  \/  ( ch  /\  ( ph \/_ ps ) ) ) )
2 idd 21 . . . 4  |-  ( -. 
ch  ->  ( ( ph  /\ 
ps )  ->  ( ph  /\  ps ) ) )
3 pm2.21 100 . . . . 5  |-  ( -. 
ch  ->  ( ch  ->  (
ph  /\  ps )
) )
43adantrd 454 . . . 4  |-  ( -. 
ch  ->  ( ( ch 
/\  ( ph \/_ ps ) )  ->  ( ph  /\  ps ) ) )
52, 4jaod 369 . . 3  |-  ( -. 
ch  ->  ( ( (
ph  /\  ps )  \/  ( ch  /\  ( ph \/_ ps ) ) )  ->  ( ph  /\ 
ps ) ) )
6 orc 374 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( ph  /\  ps )  \/  ( ch  /\  ( ph \/_ ps ) ) ) )
75, 6impbid1 194 . 2  |-  ( -. 
ch  ->  ( ( (
ph  /\  ps )  \/  ( ch  /\  ( ph \/_ ps ) ) )  <->  ( ph  /\  ps ) ) )
81, 7syl5bb 248 1  |-  ( -. 
ch  ->  (cadd ( ph ,  ps ,  ch )  <->  (
ph  /\  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   \/_wxo 1295  caddwcad 1369
This theorem is referenced by:  sadadd2lem2  12641  sadcaddlem  12648  saddisjlem  12655
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-cad 1371
  Copyright terms: Public domain W3C validator