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

Theoremxor2 1301 Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorneg1 1302 is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorneg2 1303 is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorneg 1304 is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorbi12i 1305 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremxorbi12d 1306 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

1.2.11  True and false constants

Syntaxwtru 1307 is a wff.

Syntaxwfal 1308 is a wff.

Theoremtrujust 1309 Soundness justification theorem for df-tru 1310. (Contributed by Mario Carneiro, 17-Nov-2013.)

Definitiondf-tru 1310 Definition of , a tautology. is a constant true. In this definition biid 227 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 1311 Definition of , a contradiction. is a constant false. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremtru 1312 is provable. (Contributed by Anthony Hart, 13-Oct-2010.)

Theoremfal 1313 is not provable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)

Theoremtrud 1314 Eliminate as an antecedent. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theoremtbtru 1315 If something is true, it outputs . (Contributed by Anthony Hart, 14-Aug-2011.)

Theoremnbfal 1316 If something is not true, it outputs . (Contributed by Anthony Hart, 14-Aug-2011.)

Theorembitru 1317 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)

Theorembifal 1318 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)

Theoremfalim 1319 implies anything. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)

Theoremfalimd 1320 implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theorema1tru 1321 Anything implies . (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)

Theoremdfnot 1322 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 1323 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremefald 1324 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theorempm2.21fal 1325 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 360, (disjunction aka logical inclusive 'or') df-or 359, (implies) wi 4, (not) wn 3, (logical equivalence) df-bi 177, (nand aka Sheffer stroke) df-nan 1288, and (exclusive or) df-xor 1296.

Theoremtruantru 1326 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremtruanfal 1327 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremfalantru 1328 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremfalanfal 1329 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremtruortru 1330 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtruorfal 1331 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremfalortru 1332 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremfalorfal 1333 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtruimtru 1334 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremtruimfal 1335 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremfalimtru 1336 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremfalimfal 1337 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremnottru 1338 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)

Theoremnotfal 1339 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtrubitru 1340 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtrubifal 1341 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremfalbitru 1342 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremfalbifal 1343 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtrunantru 1344 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtrunanfal 1345 A identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremfalnantru 1346 A identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremfalnanfal 1347 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremtruxortru 1348 A identity. (Contributed by David A. Wheeler, 8-May-2015.)

Theoremtruxorfal 1349 A identity. (Contributed by David A. Wheeler, 8-May-2015.)

Theoremfalxortru 1350 A identity. (Contributed by David A. Wheeler, 9-May-2015.)

Theoremfalxorfal 1351 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 1352 Virtual deduction rule e22 28443 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 1353 e12an 28500 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 1354 e23 28530 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theoremexbir 1355 Exportation implication also converting head from biconditional to conditional. This proof is exbirVD 28629 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theorem3impexp 1356 impexp 433 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)

Theorem3impexpbicom 1357 3impexp 1356 with biconditional consequent of antecedent that is commuted in consequent. Derived automatically from 3impexpVD 28632. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theorem3impexpbicomi 1358 Deduction form of 3impexpbicom 1357. Derived automatically from 3impexpbicomiVD 28634. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theoremancomsimp 1359 Closed form of ancoms 439. Derived automatically from ancomsimpVD 28641. (Contributed by Alan Sare, 31-Dec-2011.)

Theoremexp3acom3r 1360 Export and commute antecedents. (Contributed by Alan Sare, 18-Mar-2012.)

Theoremexp3acom23g 1361 Implication form of exp3acom23 1362. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theoremexp3acom23 1362 The exportation deduction exp3a 425 with commutation of the conjoined wwfs. (Contributed by Alan Sare, 22-Jul-2012.)

Theoremsimplbi2comg 1363 Implication form of simplbi2com 1364. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theoremsimplbi2com 1364 A deduction eliminating a conjunct, similar to simplbi2 608. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)

Theoremee21 1365 e21 28505 without virtual deductions. (Contributed by Alan Sare, 18-Mar-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.

Theoremee10 1366 e10 28467 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) TODO: this is frequently used; come up with better label.

Theoremee02 1367 e02 28470 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 1368 Define the half adder (triple XOR). (Contributed by Mario Carneiro, 4-Sep-2016.)

Syntaxwcad 1369 Define the half adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Definitiondf-had 1370 Define the half adder (triple XOR). (Contributed by Mario Carneiro, 4-Sep-2016.)

Definitiondf-cad 1371 Define the half adder carry, which is true when at least two arguments are true. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadbi123d 1372 Equality theorem for half adder. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadbi123d 1373 Equality theorem for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadbi123i 1374 Equality theorem for half adder. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadbi123i 1375 Equality theorem for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadass 1376 Associative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadbi 1377 The half adder is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadcoma 1378 Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadcomb 1379 Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadrot 1380 Rotation law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcador 1381 Write the adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadan 1382 Write the adder carry in conjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadnot 1383 The half adder distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadnot 1384 The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadcoma 1385 Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadcomb 1386 Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadrot 1387 Rotation law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcad1 1388 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.)

Theoremcad11 1389 If two parameters are true, the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcad0 1390 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.)

Theoremcadtru 1391 Rotation law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhad1 1392 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.)

Theoremhad0 1393 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.)

1.3  Other axiomatizations of classical propositional calculus

1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom

Theoremmeredith 1394 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 8, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 5, ax-2 6, and ax-3 7. Then from it we derive the Lukasiewicz axioms luk-1 1410, luk-2 1411, and luk-3 1412. Using these we finally re-derive our axioms as ax1 1421, ax2 1422, and ax3 1423, 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.)

Theoremaxmeredith 1395 Alias for meredith 1394 which "verify markup *" will match to ax-meredith 1396. (Contributed by NM, 21-Aug-2017.) (New usage is discouraged.)

Axiomax-meredith 1396 Theorem meredith 1394 restated as an axiom. This will allow us to ensure that the rederivation of ax1 1421, ax2 1422, and ax3 1423 below depend only on Meredith's sole axiom and not accidentally on a previous theorem above. Outside of this section, we will not make use of this axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerlem1 1397 Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.) (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerlem2 1398 Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerlem3 1399 Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerlem4 1400 Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

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