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Theorem List for Metamath Proof Explorer - 1301-1400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-xor 1301 Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. After we define the constant true  T. (df-tru 1315) and the constant false  F. (df-fal 1316), we will be able to prove these truth table values:  ( (  T.  \/_  T.  ) 
<->  F.  ) (truxortru 1354), 
( (  T.  \/_  F.  )  <->  T.  ) (truxorfal 1355),  ( (  F.  \/_  T.  )  <->  T.  ) (falxortru 1356), and  ( (  F.  \/_  F.  )  <->  F.  ) (falxorfal 1357). Contrast with  /\ (df-an 362), 
\/ (df-or 361), 
-> (wi 6), and  -/\ (df-nan 1293) . (Contributed by FL, 22-Nov-2010.)
 |-  ( ( ph \/_ ps ) 
 <->  -.  ( ph  <->  ps ) )
 
Theoremxnor 1302 Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ( ph  <->  ps )  <->  -.  ( ph \/_ ps ) )
 
Theoremxorcom 1303  \/_ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ( ph \/_ ps ) 
 <->  ( ps \/_ ph )
 )
 
Theoremxorass 1304  \/_ is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( ( ph \/_
 ps ) \/_ ch ) 
 <->  ( ph \/_ ( ps \/_ ch ) ) )
 
Theoremexcxor 1305 This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
 |-  ( ( ph \/_ ps ) 
 <->  ( ( ph  /\  -.  ps )  \/  ( -.  ph  /\  ps ) ) )
 
Theoremxor2 1306 Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ( ph \/_ ps ) 
 <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) ) )
 
Theoremxorneg1 1307  \/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ( -.  ph \/_
 ps )  <->  -.  ( ph \/_ ps ) )
 
Theoremxorneg2 1308  \/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ( ph \/_  -.  ps )  <->  -.  ( ph \/_ ps ) )
 
Theoremxorneg 1309  \/_ is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ( -.  ph \/_ 
 -.  ps )  <->  ( ph \/_ ps ) )
 
Theoremxorbi12i 1310 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph \/_ ch ) 
 <->  ( ps \/_ th )
 )
 
Theoremxorbi12d 1311 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps \/_ th )  <->  ( ch \/_ ta )
 ) )
 
1.3.11  True and false constants
 
Syntaxwtru 1312  T. is a wff.
 wff  T.
 
Syntaxwfal 1313  F. is a wff.
 wff  F.
 
Theoremtrujust 1314 Soundness justification theorem for df-tru 1315. (Contributed by Mario Carneiro, 17-Nov-2013.)
 |-  ( ( ph  <->  ph )  <->  ( ps  <->  ps ) )
 
Definitiondf-tru 1315 Definition of  T., a tautology.  T. is a constant true. In this definition biid 229 is used as an antecedent, however, any true wff, such as an axiom, can be used in its place. (Contributed by Anthony Hart, 13-Oct-2010.)
 |-  (  T.  <->  ( ph  <->  ph ) )
 
Definitiondf-fal 1316 Definition of  F., a contradiction.  F. is a constant false. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  (  F.  <->  -.  T.  )
 
Theoremtru 1317  T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
 |- 
 T.
 
Theoremtru2OLD 1318 Obsolete proof of tru 1317 as of 5-Apr-2016.  T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 T.
 
Theoremfal 1319  F. is not provable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
 |- 
 -.  F.
 
Theoremtrud 1320 Eliminate  T. as an antecedent. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  (  T.  ->  ph )   =>    |-  ph
 
Theoremtbtru 1321 If something is true, it outputs 
T.. (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  ( ph  <->  ( ph  <->  T.  ) )
 
Theoremnbfal 1322 If something is not true, it outputs  F.. (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  ( -.  ph  <->  ( ph  <->  F.  ) )
 
Theorembitru 1323 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ph   =>    |-  ( ph  <->  T.  )
 
Theorembifal 1324 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
 |- 
 -.  ph   =>    |-  ( ph  <->  F.  )
 
Theoremfalim 1325  F. implies anything. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
 |-  (  F.  ->  ph )
 
Theoremfalimd 1326  F. implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\  F.  )  ->  ps )
 
Theorema1tru 1327 Anything implies  T.. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
 |-  ( ph  ->  T.  )
 
Theoremdfnot 1328 Given falsum, we can define the negation of a wff  ph as the statement that a contradiction follows from assuming  ph. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( -.  ph  <->  ( ph  ->  F.  ) )
 
Theoreminegd 1329 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\  ps )  ->  F.  )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremefald 1330 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\  -.  ps )  ->  F.  )   =>    |-  ( ph  ->  ps )
 
Theorempm2.21fal 1331 If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  F.  )
 
1.3.12  Truth tables

Some sources define operations on true/false values using truth tables. These tables show the results of their operations for all possible combinations of true ( T.) and false ( F.). Here we show that our definitions and axioms produce equivalent results for  /\ (conjunction aka logical 'and') df-an 362,  \/ (disjunction aka logical inclusive 'or') df-or 361,  -> (implies) wi 6,  -. (not) wn 5,  <-> (logical equivalence) df-bi 179,  -/\ (nand aka Sheffer stroke) df-nan 1293, and  \/_ (exclusive or) df-xor 1301.

 
Theoremtruantru 1332 A  /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( (  T.  /\  T.  )  <->  T.  )
 
Theoremtruanfal 1333 A  /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( (  T.  /\  F.  )  <->  F.  )
 
Theoremfalantru 1334 A  /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( (  F.  /\  T.  )  <->  F.  )
 
Theoremfalanfal 1335 A  /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( (  F.  /\  F.  )  <->  F.  )
 
Theoremtruortru 1336 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  T.  \/  T.  )  <->  T.  )
 
Theoremtruorfal 1337 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( (  T.  \/  F.  )  <->  T.  )
 
Theoremfalortru 1338 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( (  F.  \/  T.  )  <->  T.  )
 
Theoremfalorfal 1339 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  F.  \/  F.  )  <->  F.  )
 
Theoremtruimtru 1340 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( (  T.  ->  T.  )  <->  T.  )
 
Theoremtruimfal 1341 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  T.  ->  F.  )  <->  F.  )
 
Theoremfalimtru 1342 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( (  F.  ->  T.  )  <->  T.  )
 
Theoremfalimfal 1343 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( (  F.  ->  F.  )  <->  T.  )
 
Theoremnottru 1344 A  -. identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( -.  T.  <->  F.  )
 
Theoremnotfal 1345 A  -. identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( -.  F.  <->  T.  )
 
Theoremtrubitru 1346 A  <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  T.  <->  T.  )  <->  T.  )
 
Theoremtrubifal 1347 A  <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  T.  <->  F.  )  <->  F.  )
 
Theoremfalbitru 1348 A  <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  F.  <->  T.  )  <->  F.  )
 
Theoremfalbifal 1349 A  <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  F.  <->  F.  )  <->  T.  )
 
Theoremtrunantru 1350 A  -/\ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  T.  -/\  T.  )  <->  F.  )
 
Theoremtrunanfal 1351 A  -/\ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  T.  -/\  F.  )  <->  T.  )
 
Theoremfalnantru 1352 A  -/\ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  F.  -/\  T.  )  <->  T.  )
 
Theoremfalnanfal 1353 A  -/\ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( (  F.  -/\  F.  )  <->  T.  )
 
Theoremtruxortru 1354 A  \/_ identity. (Contributed by David A. Wheeler, 8-May-2015.)
 |-  ( (  T.  \/_  T.  )  <->  F.  )
 
Theoremtruxorfal 1355 A  \/_ identity. (Contributed by David A. Wheeler, 8-May-2015.)
 |-  ( (  T.  \/_  F.  )  <->  T.  )
 
Theoremfalxortru 1356 A  \/_ identity. (Contributed by David A. Wheeler, 9-May-2015.)
 |-  ( (  F.  \/_  T.  )  <->  T.  )
 
Theoremfalxorfal 1357 A  \/_ identity. (Contributed by David A. Wheeler, 9-May-2015.)
 |-  ( (  F.  \/_  F.  )  <->  F.  )
 
1.3.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1
 
Theoremee22 1358 Virtual deduction rule e22 27576 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 2-May-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ch  ->  ( th  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoremee12an 1359 e12an 27633 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) TODO: this is frequently used; come up with better label.
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th ) )   &    |-  ( ( ps 
 /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ch  ->  ta ) )
 
Theoremee23 1360 e23 27663 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ta )
 ) )   &    |-  ( ch  ->  ( ta  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  et ) ) )
 
Theoremexbir 1361 Exportation implication also converting head from biconditional to conditional. This proof is exbirVD 27762 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
 |-  ( ( ( ph  /\ 
 ps )  ->  ( ch 
 <-> 
 th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch )
 ) ) )
 
Theorem3impexp 1362 impexp 435 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  ->  th )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
 
Theorem3impexpbicom 1363 3impexp 1362 with biconditional consequent of antecedent that is commuted in consequent. Derived automatically from 3impexpVD 27765. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  ->  ( th  <->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
 
Theorem3impexpbicomi 1364 Deduction form of 3impexpbicom 1363. Derived automatically from 3impexpbicomiVD 27767. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( th 
 <->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
 
Theoremancomsimp 1365 Closed form of ancoms 441. Derived automatically from ancomsimpVD 27774. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  <->  ( ( ps  /\  ph )  ->  ch ) )
 
Theoremexp3acom3r 1366 Export and commute antecedents. (Contributed by Alan Sare, 18-Mar-2012.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )
 
Theoremexp3acom23g 1367 Implication form of exp3acom23 1368. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.
 |-  ( ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) ) )
 
Theoremexp3acom23 1368 The exportation deduction exp3a 427 with commutation of the conjoined wwfs. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) )
 
Theoremsimplbi2comg 1369 Implication form of simplbi2com 1370. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.
 |-  ( ( ph  <->  ( ps  /\  ch ) )  ->  ( ch  ->  ( ps  ->  ph ) ) )
 
Theoremsimplbi2com 1370 A deduction eliminating a conjunct, similar to simplbi2 611. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ch  ->  ( ps  ->  ph ) )
 
Theoremee21 1371 e21 27638 without virtual deductions. (Contributed by Alan Sare, 18-Mar-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ( ch  ->  ( th  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoremee10 1372 e10 27600 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) TODO: this is frequently used; come up with better label.
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( ps  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  th )
 
Theoremee02 1373 e02 27603 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   =>    |-  ( ps  ->  ( ch  ->  ta ) )
 
1.3.14  Half-adders and full adders in propositional calculus

Propositional calculus deals with truth values, which can be interpreted as bits. Using this, we can define the half-adder in pure propositional calculus, and show its basic properties.

 
Syntaxwhad 1374 Define the half adder (triple XOR). (Contributed by Mario Carneiro, 4-Sep-2016.)
 wff hadd ( ph ,  ps ,  ch )
 
Syntaxwcad 1375 Define the half adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 wff cadd ( ph ,  ps ,  ch )
 
Definitiondf-had 1376 Define the half adder (triple XOR). (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (hadd ( ph ,  ps ,  ch )  <->  ( ( ph \/_
 ps ) \/_ ch ) )
 
Definitiondf-cad 1377 Define the half adder carry, which is true when at least two arguments are true. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (cadd ( ph ,  ps ,  ch )  <->  ( ( ph  /\ 
 ps )  \/  ( ch  /\  ( ph \/_ ps ) ) ) )
 
Theoremhadbi123d 1378 Equality theorem for half adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   &    |-  ( ph  ->  ( et  <->  ze ) )   =>    |-  ( ph  ->  (hadd ( ps ,  th ,  et )  <-> hadd ( ch ,  ta ,  ze ) ) )
 
Theoremcadbi123d 1379 Equality theorem for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   &    |-  ( ph  ->  ( et  <->  ze ) )   =>    |-  ( ph  ->  (cadd ( ps ,  th ,  et )  <-> cadd ( ch ,  ta ,  ze ) ) )
 
Theoremhadbi123i 1380 Equality theorem for half adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ta  <->  et )   =>    |-  (hadd ( ph ,  ch ,  ta )  <-> hadd ( ps ,  th ,  et ) )
 
Theoremcadbi123i 1381 Equality theorem for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ta  <->  et )   =>    |-  (cadd ( ph ,  ch ,  ta )  <-> cadd ( ps ,  th ,  et ) )
 
Theoremhadass 1382 Associative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (hadd ( ph ,  ps ,  ch )  <->  ( ph \/_ ( ps \/_ ch ) ) )
 
Theoremhadbi 1383 The half adder is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (hadd ( ph ,  ps ,  ch )  <->  ( ( ph  <->  ps ) 
 <->  ch ) )
 
Theoremhadcoma 1384 Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (hadd ( ph ,  ps ,  ch )  <-> hadd ( ps ,  ph ,  ch ) )
 
Theoremhadcomb 1385 Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (hadd ( ph ,  ps ,  ch )  <-> hadd ( ph ,  ch ,  ps ) )
 
Theoremhadrot 1386 Rotation law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (hadd ( ph ,  ps ,  ch )  <-> hadd ( ps ,  ch ,  ph ) )
 
Theoremcador 1387 Write the adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (cadd ( ph ,  ps ,  ch )  <->  ( ( ph  /\ 
 ps )  \/  ( ph  /\  ch )  \/  ( ps  /\  ch ) ) )
 
Theoremcadan 1388 Write the adder carry in conjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (cadd ( ph ,  ps ,  ch )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ps  \/  ch )
 ) )
 
Theoremhadnot 1389 The half adder distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( -. hadd ( ph ,  ps ,  ch )  <-> hadd ( -.  ph ,  -.  ps ,  -.  ch ) )
 
Theoremcadnot 1390 The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( -. cadd ( ph ,  ps ,  ch )  <-> cadd ( -.  ph ,  -.  ps ,  -.  ch ) )
 
Theoremcadcoma 1391 Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (cadd ( ph ,  ps ,  ch )  <-> cadd ( ps ,  ph ,  ch ) )
 
Theoremcadcomb 1392 Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (cadd ( ph ,  ps ,  ch )  <-> cadd ( ph ,  ch ,  ps ) )
 
Theoremcadrot 1393 Rotation law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (cadd ( ph ,  ps ,  ch )  <-> cadd ( ps ,  ch ,  ph ) )
 
Theoremcad1 1394 If one parameter is true, the adder carry is true exactly when at least one of the other parameters is true. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( ch  ->  (cadd ( ph ,  ps ,  ch )  <->  ( ph  \/  ps ) ) )
 
Theoremcad11 1395 If two parameters are true, the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ( ph  /\  ps )  -> cadd ( ph ,  ps ,  ch ) )
 
Theoremcad0 1396 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.)
 |-  ( -.  ch  ->  (cadd ( ph ,  ps ,  ch )  <->  ( ph  /\  ps ) ) )
 
Theoremcadtru 1397 Rotation law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |- cadd
 (  T.  ,  T.  , 
 ph )
 
Theoremhad1 1398 If the first parameter is true, the half adder is equivalent to the equality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ph  ->  (hadd ( ph ,  ps ,  ch )  <->  ( ps  <->  ch ) ) )
 
Theoremhad0 1399 If the first parameter is false, the half adder is equivalent to the XOR of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( -.  ph  ->  (hadd ( ph ,  ps ,  ch )  <->  ( ps \/_ ch ) ) )
 
1.4  Other axiomatizations of classical propositional calculus
 
1.4.1  Derive the Lukasiewicz axioms from Meredith's sole axiom
 
Theoremmeredith 1400 Carew Meredith's sole axiom for propositional calculus. This amazing formula is thought to be the shortest possible single axiom for propositional calculus with inference rule ax-mp 10, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 7, ax-2 8, and ax-3 9. Then from it we derive the Lukasiewicz axioms luk-1 1415, luk-2 1416, and luk-3 1417. Using these we finally re-derive our axioms as ax1 1426, ax2 1427, and ax3 1428, thus proving the equivalence of all three systems. C. A. Meredith, "Single Axioms for the Systems (C,N), (C,O) and (A,N) of the Two-Valued Propositional Calculus," The Journal of Computing Systems vol. 1 (1953), pp. 155-164. Meredith claimed to be close to a proof that this axiom is the shortest possible, but the proof was apparently never completed.

An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic somewhat late in life after attending talks by Lukasiewicz and produced many remarkable results such as this axiom. From his obituary: "He did logic whenever time and opportunity presented themselves, and he did it on whatever materials came to hand: in a pub, his favored pint of porter within reach, he would use the inside of cigarette packs to write proofs for logical colleagues." (Contributed by NM, 14-Dec-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Wolf Lammen, 28-May-2013.)

 |-  ( ( ( ( ( ph  ->  ps )  ->  ( -.  ch  ->  -. 
 th ) )  ->  ch )  ->  ta )  ->  ( ( ta  ->  ph )  ->  ( th  -> 
 ph ) ) )
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