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Theorem List for Metamath Proof Explorer - 1401-1500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcadcoma 1401 Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadcomb 1402 Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadrot 1403 Rotation law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcad1 1404 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 1405 If two parameters are true, the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcad0 1406 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 1407 Rotation law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhad1 1408 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 1409 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 1410 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 1426, luk-2 1427, and luk-3 1428. Using these we finally re-derive our axioms as ax1 1437, ax2 1438, and ax3 1439, 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 1411 Alias for meredith 1410 which "verify markup *" will match to ax-meredith 1412. (Contributed by NM, 21-Aug-2017.) (New usage is discouraged.)

Axiomax-meredith 1412 Theorem meredith 1410 restated as an axiom. This will allow us to ensure that the rederivation of ax1 1437, ax2 1438, and ax3 1439 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 1413 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 1414 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 1415 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 1416 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.)

Theoremmerlem5 1417 Step 11 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.)

Theoremmerlem6 1418 Step 12 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.)

Theoremmerlem7 1419 Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerlem8 1420 Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerlem9 1421 Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerlem10 1422 Step 19 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.)

Theoremmerlem11 1423 Step 20 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.)

Theoremmerlem12 1424 Step 28 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.)

Theoremmerlem13 1425 Step 35 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.)

Theoremluk-1 1426 1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremluk-2 1427 2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremluk-3 1428 3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

1.3.2  Derive the standard axioms from the Lukasiewicz axioms

Theoremluklem1 1429 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 23-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremluklem2 1430 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremluklem3 1431 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremluklem4 1432 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremluklem5 1433 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremluklem6 1434 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremluklem7 1435 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremluklem8 1436 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax1 1437 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax2 1438 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax3 1439 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

1.3.3  Derive Nicod's axiom from the standard axioms

Prove Nicod's axiom and implication and negation definitions.

Theoremnic-dfim 1440 Define implication in terms of 'nand'. Analogous to . In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition (\$a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-dfneg 1441 Define negation in terms of 'nand'. Analogous to . In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition (\$a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-mp 1442 Derive Nicod's rule of modus ponens using 'nand', from the standard one. Although the major and minor premise together also imply , this form is necessary for useful derivations from nic-ax 1444. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom (\$a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-mpALT 1443 A direct proof of nic-mp 1442. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-ax 1444 Nicod's axiom derived from the standard ones. See _Intro. to Math. Phil._ by B. Russell, p. 152. Like meredith 1410, the usual axioms can be derived from this and vice versa. Unlike meredith 1410, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g. nic-ax 1444, nic-mp 1442 is equivalent to luk-1 1426, luk-2 1427, luk-3 1428, ax-mp 8 . In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom (\$a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-axALT 1445 A direct proof of nic-ax 1444. (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom

Theoremnic-imp 1446 Inference for nic-mp 1442 using nic-ax 1444 as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-idlem1 1447 Lemma for nic-id 1449. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-idlem2 1448 Lemma for nic-id 1449. Inference used by nic-id 1449. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-id 1449 Theorem id 20 expressed with . (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-swap 1450 is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-isw1 1451 Inference version of nic-swap 1450. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-isw2 1452 Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-iimp1 1453 Inference version of nic-imp 1446 using right-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-iimp2 1454 Inference version of nic-imp 1446 using left-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-idel 1455 Inference to remove the trailing term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-ich 1456 Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-idbl 1457 Double the terms. Since doubling is the same as negation, this can be viewed as a contraposition inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-bijust 1458 For nic-* definitions, the biconditional connective is not used. Instead, definitions are made based on this form. nic-bi1 1459 and nic-bi2 1460 are used to convert the definitions into usable theorems about one side of the implication. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-bi1 1459 Inference to extract one side of an implication from a definition. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-bi2 1460 Inference to extract the other side of an implication from a 'biconditional' definition. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-stdmp 1461 Derive the standard modus ponens from nic-mp 1442. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-luk1 1462 Proof of luk-1 1426 from nic-ax 1444 and nic-mp 1442 (and definitions nic-dfim 1440 and nic-dfneg 1441). Note that the standard axioms ax-1 5, ax-2 6, and ax-3 7 are proved from the Lukasiewicz axioms by theorems ax1 1437, ax2 1438, and ax3 1439. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-luk2 1463 Proof of luk-2 1427 from nic-ax 1444 and nic-mp 1442. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnic-luk3 1464 Proof of luk-3 1428 from nic-ax 1444 and nic-mp 1442. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom

Theoremlukshef-ax1 1465 This alternative axiom for propositional calculus using the Sheffer Stroke was offered by Lukasiewicz in his Selected Works. It improves on Nicod's axiom by reducing its number of variables by one.

This axiom also uses nic-mp 1442 for its constructions.

Here, the axiom is proved as a substitution instance of nic-ax 1444. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremlukshefth1 1466 Lemma for renicax 1468. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremlukshefth2 1467 Lemma for renicax 1468. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremrenicax 1468 A rederivation of nic-ax 1444 from lukshef-ax1 1465, proving that lukshef-ax1 1465 with nic-mp 1442 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms

Theoremtbw-bijust 1469 Justification for tbw-negdf 1470. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbw-negdf 1470 The definition of negation, in terms of and . (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbw-ax1 1471 The first of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbw-ax2 1472 The second of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbw-ax3 1473 The third of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbw-ax4 1474 The fourth of four axioms in the Tarski-Bernays-Wajsberg system.

This axiom was added to the Tarski-Bernays axiom system ( see tb-ax1 26032, tb-ax2 26033, and tb-ax3 26034) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbwsyl 1475 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbwlem1 1476 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbwlem2 1477 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbwlem3 1478 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbwlem4 1479 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbwlem5 1480 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremre1luk1 1481 luk-1 1426 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremre1luk2 1482 luk-2 1427 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremre1luk3 1483 luk-3 1428 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1481 and re1luk2 1482 proves that tbw-ax1 1471, tbw-ax2 1472, tbw-ax3 1473, and tbw-ax4 1474, with ax-mp 8 can be used as a complete axiom system for all of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom

Theoremmerco1 1484 A single axiom for propositional calculus offered by Meredith.

This axiom is worthy of note, due to it having only 19 symbols, not counting parentheses. The more well-known meredith 1410 has 21 symbols, sans parentheses.

See merco2 1507 for another axiom of equal length. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem1 1485 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremretbwax4 1486 tbw-ax4 1474 rederived from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremretbwax2 1487 tbw-ax2 1472 rederived from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem2 1488 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem3 1489 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem4 1490 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem5 1491 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem6 1492 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem7 1493 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremretbwax3 1494 tbw-ax3 1473 rederived from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem8 1495 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem9 1496 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem10 1497 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem11 1498 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem12 1499 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmerco1lem13 1500 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1484. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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