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Theorem List for Metamath Proof Explorer - 1301-1400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnanbi1 1301 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)

Theoremnanbi2 1302 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)

Theoremnanbi12 1303 Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)

Theoremnanbi1i 1304 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)

Theoremnanbi2i 1305 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)

Theoremnanbi12i 1306 Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)

Theoremnanbi1d 1307 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)

Theoremnanbi2d 1308 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)

Theoremnanbi12d 1309 Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018.)

1.2.10  Logical 'xor'

Syntaxwxo 1310 Extend wff definition to include exclusive disjunction ('xor').

Definitiondf-xor 1311 Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. After we define the constant true (df-tru 1325) and the constant false (df-fal 1326), we will be able to prove these truth table values: (truxortru 1364), (truxorfal 1365), (falxortru 1366), and (falxorfal 1367). Contrast with (df-an 361), (df-or 360), (wi 4), and (df-nan 1294) . (Contributed by FL, 22-Nov-2010.)

Theoremxnor 1312 Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorcom 1313 is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorass 1314 is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremexcxor 1315 This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)

Theoremxor2 1316 Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorneg1 1317 is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorneg2 1318 is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorneg 1319 is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorbi12i 1320 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorbi12d 1321 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

1.2.11  True and false constants

Syntaxwtru 1322 is a wff.

Syntaxwfal 1323 is a wff.

Theoremtrujust 1324 Soundness justification theorem for df-tru 1325. (Contributed by Mario Carneiro, 17-Nov-2013.)

Definitiondf-tru 1325 Definition of , a tautology. is a constant true. In this definition biid 228 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.)

Definitiondf-fal 1326 Definition of , a contradiction. is a constant false. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremtru 1327 is provable. (Contributed by Anthony Hart, 13-Oct-2010.)

Theoremfal 1328 is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)

Theoremtrud 1329 Eliminate as an antecedent. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theoremtbtru 1330 If something is true, it outputs . (Contributed by Anthony Hart, 14-Aug-2011.)

Theoremnbfal 1331 If something is not true, it outputs . (Contributed by Anthony Hart, 14-Aug-2011.)

Theorembitru 1332 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)

Theorembifal 1333 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)

Theoremfalim 1334 implies anything. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)

Theoremfalimd 1335 implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theorema1tru 1336 Anything implies . (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)

Theoremtruan 1337 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.)

Theoremdfnot 1338 Given falsum, we can define the negation of a wff as the statement that a contradiction follows from assuming . (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoreminegd 1339 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremefald 1340 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theorempm2.21fal 1341 If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)

1.2.12  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 () and false (). Here we show that our definitions and axioms produce equivalent results for (conjunction aka logical 'and') df-an 361, (disjunction aka logical inclusive 'or') df-or 360, (implies) wi 4, (not) wn 3, (logical equivalence) df-bi 178, (nand aka Sheffer stroke) df-nan 1294, and (exclusive or) df-xor 1311.

Theoremtruantru 1342 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremtruanfal 1343 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremfalantru 1344 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremfalanfal 1345 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremtruortru 1346 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtruorfal 1347 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremfalortru 1348 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremfalorfal 1349 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtruimtru 1350 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremtruimfal 1351 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremfalimtru 1352 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremfalimfal 1353 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremnottru 1354 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremnotfal 1355 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtrubitru 1356 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtrubifal 1357 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremfalbitru 1358 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremfalbifal 1359 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtrunantru 1360 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtrunanfal 1361 A identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremfalnantru 1362 A identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremfalnanfal 1363 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtruxortru 1364 A identity. (Contributed by David A. Wheeler, 8-May-2015.)

Theoremtruxorfal 1365 A identity. (Contributed by David A. Wheeler, 8-May-2015.)

Theoremfalxortru 1366 A identity. (Contributed by David A. Wheeler, 9-May-2015.)

Theoremfalxorfal 1367 A identity. (Contributed by David A. Wheeler, 9-May-2015.)

1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1

Theoremee22 1368 Virtual deduction rule e22 28481 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.

Theoremee12an 1369 e12an 28546 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.

Theoremee23 1370 e23 28576 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theoremexbir 1371 Exportation implication also converting head from biconditional to conditional. This proof is exbirVD 28674 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theorem3impexp 1372 impexp 434 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)

Theorem3impexpbicom 1373 3impexp 1372 with biconditional consequent of antecedent that is commuted in consequent. Derived automatically from 3impexpVD 28677. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theorem3impexpbicomi 1374 Deduction form of 3impexpbicom 1373. Derived automatically from 3impexpbicomiVD 28679. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theoremancomsimp 1375 Closed form of ancoms 440. Derived automatically from ancomsimpVD 28686. (Contributed by Alan Sare, 31-Dec-2011.)

Theoremexp3acom3r 1376 Export and commute antecedents. (Contributed by Alan Sare, 18-Mar-2012.)

Theoremexp3acom23g 1377 Implication form of exp3acom23 1378. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theoremexp3acom23 1378 The exportation deduction exp3a 426 with commutation of the conjoined wwfs. (Contributed by Alan Sare, 22-Jul-2012.)

Theoremsimplbi2comg 1379 Implication form of simplbi2com 1380. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theoremsimplbi2com 1380 A deduction eliminating a conjunct, similar to simplbi2 609. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)

Theoremee21 1381 e21 28551 without virtual deductions. (Contributed by Alan Sare, 18-Mar-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theoremee10 1382 e10 28504 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) TODO: this is frequently used; come up with better label.

Theoremee02 1383 e02 28507 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.

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 1384 Define the half adder (triple XOR). (Contributed by Mario Carneiro, 4-Sep-2016.)

Syntaxwcad 1385 Define the half adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Definitiondf-had 1386 Define the half adder (triple XOR). (Contributed by Mario Carneiro, 4-Sep-2016.)

Definitiondf-cad 1387 Define the half adder carry, which is true when at least two arguments are true. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadbi123d 1388 Equality theorem for half adder. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadbi123d 1389 Equality theorem for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadbi123i 1390 Equality theorem for half adder. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadbi123i 1391 Equality theorem for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadass 1392 Associative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadbi 1393 The half adder is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadcoma 1394 Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadcomb 1395 Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadrot 1396 Rotation law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcador 1397 Write the adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.)