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Theorem List for Metamath Proof Explorer - 1401-1500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmerlem5 1401 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 1402 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 1403 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 1404 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 1405 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 1406 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 1407 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 1408 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 1409 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 1410 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 1411 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 1412 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 1413 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 1414 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 1415 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 1416 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 1417 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 1418 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 1419 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 1420 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 1421 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 1422 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 1423 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 1424 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 1425 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 1426 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 1428. 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 1427 A direct proof of nic-mp 1426. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  -/\  ( ch  -/\  ps ) )   =>    |-  ps
 
Theoremnic-ax 1428 Nicod's axiom derived from the standard ones. See _Intro. to Math. Phil._ by B. Russell, p. 152. Like meredith 1394, the usual axioms can be derived from this and vice versa. Unlike meredith 1394, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g.  { nic-ax 1428, nic-mp 1426  } is equivalent to  { luk-1 1410, luk-2 1411, luk-3 1412, 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 1429 A direct proof of nic-ax 1428. (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 1430 Inference for nic-mp 1426 using nic-ax 1428 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 1431 Lemma for nic-id 1433. (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 1432 Lemma for nic-id 1433. Inference used by nic-id 1433. (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 1433 Theorem id 19 expressed with  -/\. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ta  -/\  ( ta  -/\  ta ) )
 
Theoremnic-swap 1434  -/\ 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 1435 Inference version of nic-swap 1434. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( th  -/\  ph )   =>    |-  ( ph  -/\  th )
 
Theoremnic-isw2 1436 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 1437 Inference version of nic-imp 1430 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 1438 Inference version of nic-imp 1430 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 1439 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 1440 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 1441 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 1442 For nic-* definitions, the biconditional connective is not used. Instead, definitions are made based on this form. nic-bi1 1443 and nic-bi2 1444 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 1443 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 1444 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 1445 Derive the standard modus ponens from nic-mp 1426. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  ps
 
Theoremnic-luk1 1446 Proof of luk-1 1410 from nic-ax 1428 and nic-mp 1426 (and definitions nic-dfim 1424 and nic-dfneg 1425). Note that the standard axioms ax-1 5, ax-2 6, and ax-3 7 are proved from the Lukasiewicz axioms by theorems ax1 1421, ax2 1422, and ax3 1423. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theoremnic-luk2 1447 Proof of luk-2 1411 from nic-ax 1428 and nic-mp 1426. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( -.  ph  -> 
 ph )  ->  ph )
 
Theoremnic-luk3 1448 Proof of luk-3 1412 from nic-ax 1428 and nic-mp 1426. (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 1449 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 1426 for its constructions.

Here, the axiom is proved as a substitution instance of nic-ax 1428. (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 1450 Lemma for renicax 1452. (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 1451 Lemma for renicax 1452. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ta  -/\  th )  -/\  ( ( th  -/\  ta )  -/\  ( th  -/\  ta )
 ) )
 
Theoremrenicax 1452 A rederivation of nic-ax 1428 from lukshef-ax1 1449, proving that lukshef-ax1 1449 with nic-mp 1426 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 1453 Justification for tbw-negdf 1454. (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 1454 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 1455 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 1456 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 1457 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 1458 The fourth of four axioms in the Tarski-Bernays-Wajsberg system.

This axiom was added to the Tarski-Bernays axiom system ( see tb-ax1 24817, tb-ax2 24818, and tb-ax3 24819) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  (  F.  ->  ph )
 
Theoremtbwsyl 1459 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 1460 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 1461 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 1462 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 1463 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 1464 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 1465 luk-1 1410 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 1466 luk-2 1411 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 1467 luk-3 1412 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1465 and re1luk2 1466 proves that tbw-ax1 1455, tbw-ax2 1456, tbw-ax3 1457, and tbw-ax4 1458, 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 1468 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 1394 has 21 symbols, sans parentheses.

See merco2 1491 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 1469 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  (  F.  ->  ch ) )
 
Theoremretbwax4 1470 tbw-ax4 1458 rederived from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  F.  ->  ph )
 
Theoremretbwax2 1471 tbw-ax2 1456 rederived from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremmerco1lem2 1472 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( ( ( ps 
 ->  ta )  ->  ( ph  ->  F.  ) )  ->  ch ) )
 
Theoremmerco1lem3 1473 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  ->  ( ch  ->  ph )
 )
 
Theoremmerco1lem4 1474 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( ps  ->  ch )
 )
 
Theoremmerco1lem5 1475 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( (
 ph  ->  F.  )  ->  ch )  ->  ta )  ->  ( ph  ->  ta )
 )
 
Theoremmerco1lem6 1476 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  ->  ( ch  ->  ( ph  ->  ps ) ) )
 
Theoremmerco1lem7 1477 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  (
 ( ( ps  ->  ch )  ->  ps )  ->  ps ) )
 
Theoremretbwax3 1478 tbw-ax3 1457 rederived from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ph )  -> 
 ph )
 
Theoremmerco1lem8 1479 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  (
 ( ps  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ch )
 ) )
 
Theoremmerco1lem9 1480 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremmerco1lem10 1481 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ( ( ph  ->  ps )  ->  ch )  ->  ( ta  ->  ch ) )  ->  ph )  ->  ( th  -> 
 ph ) )
 
Theoremmerco1lem11 1482 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( (
 ( ch  ->  ( ph  ->  ta ) )  ->  F.  )  ->  ps )
 )
 
Theoremmerco1lem12 1483 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( (
 ( ch  ->  ( ph  ->  ta ) )  ->  ph )  ->  ps )
 )
 
Theoremmerco1lem13 1484 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( (
 ph  ->  ps )  ->  ( ch  ->  ps ) )  ->  ta )  ->  ( ph  ->  ta ) )
 
Theoremmerco1lem14 1485 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( (
 ph  ->  ps )  ->  ps )  ->  ch )  ->  ( ph  ->  ch ) )
 
Theoremmerco1lem15 1486 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ph  ->  ( ch  ->  ps )
 ) )
 
Theoremmerco1lem16 1487 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ( ps  ->  ch )
 )  ->  ta )  ->  ( ( ph  ->  ch )  ->  ta )
 )
 
Theoremmerco1lem17 1488 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ( ( ph  ->  ps )  -> 
 ph )  ->  ch )  ->  ta )  ->  (
 ( ph  ->  ch )  ->  ta ) )
 
Theoremmerco1lem18 1489 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ( ps  ->  ph )  ->  ( ps  ->  ch ) ) )
 
Theoremretbwax1 1490 tbw-ax1 1455 rederived from merco1 1468.

This theorem, along with retbwax2 1471, retbwax3 1478, and retbwax4 1470, shows that merco1 1468 with ax-mp 8 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom
 
Theoremmerco2 1491 A single axiom for propositional calculus offered by Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1468. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( ( ( ph  ->  ps )  ->  (
 (  F.  ->  ch )  ->  th ) )  ->  ( ( th  ->  ph )  ->  ( ta  ->  ( et  ->  ph )
 ) ) )
 
Theoremmercolem1 1492 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( ps  ->  ( th  ->  ch ) ) )
 
Theoremmercolem2 1493 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  ->  ph )  ->  ( ch  ->  ( th  ->  ph ) ) )
 
Theoremmercolem3 1494 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ps  ->  ch )  ->  ( ps  ->  ( ph  ->  ch )
 ) )
 
Theoremmercolem4 1495 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( th  ->  ( et  ->  ph ) ) 
 ->  ( ( ( th  ->  ch )  ->  ph )  ->  ( ta  ->  ( et  ->  ph ) ) ) )
 
Theoremmercolem5 1496 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( th  ->  (
 ( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) )
 
Theoremmercolem6 1497 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) ) 
 ->  ( ps  ->  ( ph  ->  ch ) ) )
 
Theoremmercolem7 1498 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( (
 ( ph  ->  ch )  ->  ( th  ->  ps )
 )  ->  ( th  ->  ps ) ) )
 
Theoremmercolem8 1499 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ( ph  ->  ch ) )  ->  ( ta  ->  ( th  ->  (
 ph  ->  ch ) ) ) ) )
 
Theoremre1tbw1 1500 tbw-ax1 1455 rederived from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
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