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Theorem List for Metamath Proof Explorer - 1401-1500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcador 1401 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 1402 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 1403 The half adder distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( -. hadd ( ph ,  ps ,  ch )  <-> hadd ( -.  ph ,  -.  ps ,  -.  ch ) )
 
Theoremcadnot 1404 The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( -. cadd ( ph ,  ps ,  ch )  <-> cadd ( -.  ph ,  -.  ps ,  -.  ch ) )
 
Theoremcadcoma 1405 Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (cadd ( ph ,  ps ,  ch )  <-> cadd ( ps ,  ph ,  ch ) )
 
Theoremcadcomb 1406 Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (cadd ( ph ,  ps ,  ch )  <-> cadd ( ph ,  ch ,  ps ) )
 
Theoremcadrot 1407 Rotation law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  (cadd ( ph ,  ps ,  ch )  <-> cadd ( ps ,  ch ,  ph ) )
 
Theoremcad1 1408 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 1409 If two parameters are true, the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ( ph  /\  ps )  -> cadd ( ph ,  ps ,  ch ) )
 
Theoremcad0 1410 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 1411 Rotation law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |- cadd
 (  T.  ,  T.  , 
 ph )
 
Theoremhad1 1412 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 1413 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 ) ) )
 
1.3  Other axiomatizations of classical propositional calculus
 
1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom
 
Theoremmeredith 1414 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 1430, luk-2 1431, and luk-3 1432. Using these we finally re-derive our axioms as ax1 1441, ax2 1442, and ax3 1443, 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 ) ) )
 
Theoremaxmeredith 1415 Alias for meredith 1414 which "verify markup *" will match to ax-meredith 1416. (Contributed by NM, 21-Aug-2017.) (New usage is discouraged.)
 |-  ( ( ( ( ( ph  ->  ps )  ->  ( -.  ch  ->  -. 
 th ) )  ->  ch )  ->  ta )  ->  ( ( ta  ->  ph )  ->  ( th  -> 
 ph ) ) )
 
Axiomax-meredith 1416 Theorem meredith 1414 restated as an axiom. This will allow us to ensure that the rederivation of ax1 1441, ax2 1442, and ax3 1443 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.)
 |-  ( ( ( ( ( ph  ->  ps )  ->  ( -.  ch  ->  -. 
 th ) )  ->  ch )  ->  ta )  ->  ( ( ta  ->  ph )  ->  ( th  -> 
 ph ) ) )
 
Theoremmerlem1 1417 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.)
 |-  ( ( ( ch 
 ->  ( -.  ph  ->  ps ) )  ->  ta )  ->  ( ph  ->  ta )
 )
 
Theoremmerlem2 1418 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.)
 |-  ( ( ( ph  -> 
 ph )  ->  ch )  ->  ( th  ->  ch )
 )
 
Theoremmerlem3 1419 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.)
 |-  ( ( ( ps 
 ->  ch )  ->  ph )  ->  ( ch  ->  ph )
 )
 
Theoremmerlem4 1420 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.)
 |-  ( ta  ->  (
 ( ta  ->  ph )  ->  ( th  ->  ph )
 ) )
 
Theoremmerlem5 1421 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.)
 |-  ( ( ph  ->  ps )  ->  ( -.  -.  ph  ->  ps ) )
 
Theoremmerlem6 1422 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.)
 |-  ( ch  ->  (
 ( ( ps  ->  ch )  ->  ph )  ->  ( th  ->  ph ) ) )
 
Theoremmerlem7 1423 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.)
 |-  ( ph  ->  (
 ( ( ps  ->  ch )  ->  th )  ->  ( ( ( ch 
 ->  ta )  ->  ( -.  th  ->  -.  ps )
 )  ->  th )
 ) )
 
Theoremmerlem8 1424 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.)
 |-  ( ( ( ps 
 ->  ch )  ->  th )  ->  ( ( ( ch 
 ->  ta )  ->  ( -.  th  ->  -.  ps )
 )  ->  th )
 )
 
Theoremmerlem9 1425 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.)
 |-  ( ( ( ph  ->  ps )  ->  ( ch  ->  ( th  ->  ( ps  ->  ta )
 ) ) )  ->  ( et  ->  ( ch 
 ->  ( th  ->  ( ps  ->  ta ) ) ) ) )
 
Theoremmerlem10 1426 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.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  ->  ( th  ->  ( ph  ->  ps ) ) )
 
Theoremmerlem11 1427 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.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremmerlem12 1428 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.)
 |-  ( ( ( th  ->  ( -.  -.  ch  ->  ch ) )  ->  ph )  ->  ph )
 
Theoremmerlem13 1429 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.)
 |-  ( ( ph  ->  ps )  ->  ( (
 ( th  ->  ( -. 
 -.  ch  ->  ch )
 )  ->  -.  -.  ph )  ->  ps ) )
 
Theoremluk-1 1430 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.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theoremluk-2 1431 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.)
 |-  ( ( -.  ph  -> 
 ph )  ->  ph )
 
Theoremluk-3 1432 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.)
 |-  ( ph  ->  ( -.  ph  ->  ps )
 )
 
1.3.2  Derive the standard axioms from the Lukasiewicz axioms
 
Theoremluklem1 1433 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 23-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremluklem2 1434 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  -. 
 ps )  ->  (
 ( ( ph  ->  ch )  ->  th )  ->  ( ps  ->  th )
 ) )
 
Theoremluklem3 1435 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  (
 ( ( -.  ph  ->  ps )  ->  ch )  ->  ( th  ->  ch )
 ) )
 
Theoremluklem4 1436 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ( -.  ph  ->  ph )  -> 
 ph )  ->  ps )  ->  ps )
 
Theoremluklem5 1437 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremluklem6 1438 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremluklem7 1439 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ps  ->  ( ph  ->  ch )
 ) )
 
Theoremluklem8 1440 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ( ch  ->  ph )  ->  ( ch  ->  ps ) ) )
 
Theoremax1 1441 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremax2 1442 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
 
Theoremax3 1443 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( -.  ph  ->  -.  ps )  ->  ( ps  ->  ph ) )
 
1.3.3  Derive Nicod's axiom from the standard axioms

Prove Nicod's axiom and implication and negation definitions.

 
Theoremnic-dfim 1444 Define implication in terms of 'nand'. Analogous to  ( ( ph  -/\  ( ps  -/\  ps ) )  <->  ( ph  ->  ps ) ). 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.)
 |-  ( ( ( ph  -/\  ( ps  -/\  ps )
 )  -/\  ( ph  ->  ps ) )  -/\  (
 ( ( ph  -/\  ( ps  -/\  ps ) ) 
 -/\  ( ph  -/\  ( ps  -/\  ps ) ) )  -/\  ( ( ph  ->  ps )  -/\  ( ph  ->  ps ) ) ) )
 
Theoremnic-dfneg 1445 Define negation in terms of 'nand'. Analogous to  ( ( ph  -/\  ph )  <->  -.  ph ). 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.)
 |-  ( ( ( ph  -/\  ph )  -/\  -.  ph )  -/\  ( ( (
 ph  -/\  ph )  -/\  ( ph  -/\  ph ) )  -/\  ( -.  ph  -/\  -.  ph ) ) )
 
Theoremnic-mp 1446 Derive Nicod's rule of modus ponens using 'nand', from the standard one. Although the major and minor premise together also imply  ch, this form is necessary for useful derivations from nic-ax 1448. 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.)
 |-  ph   &    |-  ( ph  -/\  ( ch  -/\  ps ) )   =>    |-  ps
 
Theoremnic-mpALT 1447 A direct proof of nic-mp 1446. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  -/\  ( ch  -/\  ps ) )   =>    |-  ps
 
Theoremnic-ax 1448 Nicod's axiom derived from the standard ones. See _Intro. to Math. Phil._ by B. Russell, p. 152. Like meredith 1414, the usual axioms can be derived from this and vice versa. Unlike meredith 1414, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g.  { nic-ax 1448, nic-mp 1446  } is equivalent to  { luk-1 1430, luk-2 1431, luk-3 1432, 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.)
 |-  ( ( ph  -/\  ( ch  -/\  ps ) ) 
 -/\  ( ( ta  -/\  ( ta  -/\  ta )
 )  -/\  ( ( th  -/\ 
 ch )  -/\  (
 ( ph  -/\  th )  -/\  ( ph  -/\  th )
 ) ) ) )
 
Theoremnic-axALT 1449 A direct proof of nic-ax 1448. (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  -/\  ( ch  -/\  ps ) ) 
 -/\  ( ( ta  -/\  ( ta  -/\  ta )
 )  -/\  ( ( th  -/\ 
 ch )  -/\  (
 ( ph  -/\  th )  -/\  ( ph  -/\  th )
 ) ) ) )
 
1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom
 
Theoremnic-imp 1450 Inference for nic-mp 1446 using nic-ax 1448 as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  -/\  ( ch  -/\  ps ) )   =>    |-  ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
 ) )
 
Theoremnic-idlem1 1451 Lemma for nic-id 1453. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( th  -/\  ( ta  -/\  ( ta  -/\  ta )
 ) )  -/\  (
 ( ( ph  -/\  ( ch  -/\  ps ) ) 
 -/\  th )  -/\  (
 ( ph  -/\  ( ch  -/\  ps ) )  -/\  th ) ) )
 
Theoremnic-idlem2 1452 Lemma for nic-id 1453. Inference used by nic-id 1453. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( et  -/\  (
 ( ph  -/\  ( ch  -/\  ps ) )  -/\  th ) )   =>    |-  ( ( th  -/\  ( ta  -/\  ( ta  -/\  ta )
 ) )  -/\  et )
 
Theoremnic-id 1453 Theorem id 21 expressed with  -/\. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ta  -/\  ( ta  -/\  ta ) )
 
Theoremnic-swap 1454  -/\ is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( th  -/\  ph )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
 ) )
 
Theoremnic-isw1 1455 Inference version of nic-swap 1454. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( th  -/\  ph )   =>    |-  ( ph  -/\  th )
 
Theoremnic-isw2 1456 Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ps  -/\  ( th  -/\  ph ) )   =>    |-  ( ps  -/\  ( ph  -/\  th ) )
 
Theoremnic-iimp1 1457 Inference version of nic-imp 1450 using right-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  -/\  ( ch  -/\  ps ) )   &    |-  ( th  -/\  ch )   =>    |-  ( th  -/\  ph )
 
Theoremnic-iimp2 1458 Inference version of nic-imp 1450 using left-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  -/\  ps )  -/\  ( ch  -/\  ch )
 )   &    |-  ( th  -/\  ph )   =>    |-  ( th  -/\  ( ch  -/\  ch )
 )
 
Theoremnic-idel 1459 Inference to remove the trailing term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  -/\  ( ch  -/\  ps ) )   =>    |-  ( ph  -/\  ( ch  -/\ 
 ch ) )
 
Theoremnic-ich 1460 Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  -/\  ( ps  -/\  ps ) )   &    |-  ( ps  -/\  ( ch  -/\ 
 ch ) )   =>    |-  ( ph  -/\  ( ch  -/\  ch ) )
 
Theoremnic-idbl 1461 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.)
 |-  ( ph  -/\  ( ps  -/\  ps ) )   =>    |-  ( ( ps  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ph  -/\  ph )
 ) )
 
Theoremnic-bijust 1462 For nic-* definitions, the biconditional connective is not used. Instead, definitions are made based on this form. nic-bi1 1463 and nic-bi2 1464 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.)
 |-  ( ( ta  -/\  ta )  -/\  ( ( ta  -/\  ta )  -/\  ( ta  -/\  ta )
 ) )
 
Theoremnic-bi1 1463 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.)
 |-  ( ( ph  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
 ) )   =>    |-  ( ph  -/\  ( ps  -/\  ps ) )
 
Theoremnic-bi2 1464 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.)
 |-  ( ( ph  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
 ) )   =>    |-  ( ps  -/\  ( ph  -/\  ph ) )
 
Theoremnic-stdmp 1465 Derive the standard modus ponens from nic-mp 1446. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  ps
 
Theoremnic-luk1 1466 Proof of luk-1 1430 from nic-ax 1448 and nic-mp 1446 (and definitions nic-dfim 1444 and nic-dfneg 1445). Note that the standard axioms ax-1 5, ax-2 6, and ax-3 7 are proved from the Lukasiewicz axioms by theorems ax1 1441, ax2 1442, and ax3 1443. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theoremnic-luk2 1467 Proof of luk-2 1431 from nic-ax 1448 and nic-mp 1446. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( -.  ph  -> 
 ph )  ->  ph )
 
Theoremnic-luk3 1468 Proof of luk-3 1432 from nic-ax 1448 and nic-mp 1446. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( -.  ph  ->  ps )
 )
 
1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom
 
Theoremlukshef-ax1 1469 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 1446 for its constructions.

Here, the axiom is proved as a substitution instance of nic-ax 1448. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( ( ph  -/\  ( ch  -/\  ps ) ) 
 -/\  ( ( th  -/\  ( th  -/\  th )
 )  -/\  ( ( th  -/\ 
 ch )  -/\  (
 ( ph  -/\  th )  -/\  ( ph  -/\  th )
 ) ) ) )
 
Theoremlukshefth1 1470 Lemma for renicax 1472. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
 ) )  -/\  ( th  -/\  ( th  -/\  th )
 ) )  -/\  ( ph  -/\  ( ps  -/\  ch )
 ) )
 
Theoremlukshefth2 1471 Lemma for renicax 1472. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ta  -/\  th )  -/\  ( ( th  -/\  ta )  -/\  ( th  -/\  ta )
 ) )
 
Theoremrenicax 1472 A rederivation of nic-ax 1448 from lukshef-ax1 1469, proving that lukshef-ax1 1469 with nic-mp 1446 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.)
 |-  ( ( ph  -/\  ( ch  -/\  ps ) ) 
 -/\  ( ( ta  -/\  ( ta  -/\  ta )
 )  -/\  ( ( th  -/\ 
 ch )  -/\  (
 ( ph  -/\  th )  -/\  ( ph  -/\  th )
 ) ) ) )
 
1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms
 
Theoremtbw-bijust 1473 Justification for tbw-negdf 1474. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  <->  ps )  <->  ( ( (
 ph  ->  ps )  ->  (
 ( ps  ->  ph )  ->  F.  ) )  ->  F.  ) )
 
Theoremtbw-negdf 1474 The definition of negation, in terms of  -> and 
F.. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( -.  ph  ->  ( ph  ->  F.  ) )  ->  (
 ( ( ph  ->  F.  )  ->  -.  ph )  ->  F.  ) )  ->  F.  )
 
Theoremtbw-ax1 1475 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.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theoremtbw-ax2 1476 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.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremtbw-ax3 1477 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.)
 |-  ( ( ( ph  ->  ps )  ->  ph )  -> 
 ph )
 
Theoremtbw-ax4 1478 The fourth of four axioms in the Tarski-Bernays-Wajsberg system.

This axiom was added to the Tarski-Bernays axiom system ( see tb-ax1 26130, tb-ax2 26131, and tb-ax3 26132) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  (  F.  ->  ph )
 
Theoremtbwsyl 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.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremtbwlem1 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.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ps  ->  ( ph  ->  ch )
 ) )
 
Theoremtbwlem2 1481 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.)
 |-  ( ( ph  ->  ( ps  ->  F.  )
 )  ->  ( (
 ( ph  ->  ch )  ->  th )  ->  ( ps  ->  th ) ) )
 
Theoremtbwlem3 1482 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.)
 |-  ( ( ( ( ( ph  ->  F.  )  -> 
 ph )  ->  ph )  ->  ps )  ->  ps )
 
Theoremtbwlem4 1483 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.)
 |-  ( ( ( ph  ->  F.  )  ->  ps )  ->  ( ( ps  ->  F.  )  ->  ph ) )
 
Theoremtbwlem5 1484 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.)
 |-  ( ( ( ph  ->  ( ps  ->  F.  )
 )  ->  F.  )  -> 
 ph )
 
Theoremre1luk1 1485 luk-1 1430 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theoremre1luk2 1486 luk-2 1431 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( -.  ph  -> 
 ph )  ->  ph )
 
Theoremre1luk3 1487 luk-3 1432 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1485 and re1luk2 1486 proves that tbw-ax1 1475, tbw-ax2 1476, tbw-ax3 1477, and tbw-ax4 1478, 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.)

 |-  ( ph  ->  ( -.  ph  ->  ps )
 )
 
1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom
 
Theoremmerco1 1488 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 1414 has 21 symbols, sans parentheses.

See merco2 1511 for another axiom of equal length. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( ( ( ( ( ph  ->  ps )  ->  ( ch  ->  F.  )
 )  ->  th )  ->  ta )  ->  (
 ( ta  ->  ph )  ->  ( ch  ->  ph )
 ) )
 
Theoremmerco1lem1 1489 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  (  F.  ->  ch ) )
 
Theoremretbwax4 1490 tbw-ax4 1478 rederived from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  F.  ->  ph )
 
Theoremretbwax2 1491 tbw-ax2 1476 rederived from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremmerco1lem2 1492 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( ( ( ps 
 ->  ta )  ->  ( ph  ->  F.  ) )  ->  ch ) )
 
Theoremmerco1lem3 1493 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  ->  ( ch  ->  ph )
 )
 
Theoremmerco1lem4 1494 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( ps  ->  ch )
 )
 
Theoremmerco1lem5 1495 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( (
 ph  ->  F.  )  ->  ch )  ->  ta )  ->  ( ph  ->  ta )
 )
 
Theoremmerco1lem6 1496 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  ->  ( ch  ->  ( ph  ->  ps ) ) )
 
Theoremmerco1lem7 1497 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  (
 ( ( ps  ->  ch )  ->  ps )  ->  ps ) )
 
Theoremretbwax3 1498 tbw-ax3 1477 rederived from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ph )  -> 
 ph )
 
Theoremmerco1lem8 1499 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1488. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  (
 ( ps  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ch )
 ) )
 
Theoremmerco1lem9 1500 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1488. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  ->  ( ph  ->  ps )
 )
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