HomeHome Metamath Proof Explorer
Theorem List (p. 17 of 429)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27903)
  Hilbert Space Explorer  Hilbert Space Explorer
(27904-29428)
  Users' Mathboxes  Users' Mathboxes
(29429-42879)
 

Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremminimp-sylsimp 1601 Derivation of sylsimp (jarr 106) from ax-mp 5 and minimp 1600. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))
 
Theoremminimp-ax1 1602 Derivation of ax-1 6 from ax-mp 5 and minimp 1600. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
Theoremminimp-ax2c 1603 Derivation of a commuted form of ax-2 7 from ax-mp 5 and minimp 1600. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
 
Theoremminimp-ax2 1604 Derivation of ax-2 7 from ax-mp 5 and minimp 1600. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremminimp-pm2.43 1605 Derivation of pm2.43 56 (also called "hilbert" or W) from ax-mp 5 and minimp 1600. It uses the classical derivation from ax-1 6 and ax-2 7 written DD22D21 in D-notation (see head comment for an explanation) and shortens the proof using mp2 9 (which only requires ax-mp 5). (Contributed by BJ, 31-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))
 
1.3.2  Derive the Lukasiewicz axioms from Meredith's sole axiom
 
Theoremmeredith 1606 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 5, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 6, ax-2 7, and ax-3 8. Then from it we derive the Lukasiewicz axioms luk-1 1620, luk-2 1621, and luk-3 1622. Using these we finally rederive our axioms as ax1 1631, ax2 1632, and ax3 1633, 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.)

(((((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → 𝜏) → ((𝜏𝜑) → (𝜃𝜑)))
 
Theoremmerlem1 1607 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.)
(((𝜒 → (¬ 𝜑𝜓)) → 𝜏) → (𝜑𝜏))
 
Theoremmerlem2 1608 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.)
(((𝜑𝜑) → 𝜒) → (𝜃𝜒))
 
Theoremmerlem3 1609 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.)
(((𝜓𝜒) → 𝜑) → (𝜒𝜑))
 
Theoremmerlem4 1610 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.)
(𝜏 → ((𝜏𝜑) → (𝜃𝜑)))
 
Theoremmerlem5 1611 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.)
((𝜑𝜓) → (¬ ¬ 𝜑𝜓))
 
Theoremmerlem6 1612 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.)
(𝜒 → (((𝜓𝜒) → 𝜑) → (𝜃𝜑)))
 
Theoremmerlem7 1613 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.)
(𝜑 → (((𝜓𝜒) → 𝜃) → (((𝜒𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)))
 
Theoremmerlem8 1614 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.)
(((𝜓𝜒) → 𝜃) → (((𝜒𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))
 
Theoremmerlem9 1615 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.)
(((𝜑𝜓) → (𝜒 → (𝜃 → (𝜓𝜏)))) → (𝜂 → (𝜒 → (𝜃 → (𝜓𝜏)))))
 
Theoremmerlem10 1616 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.)
((𝜑 → (𝜑𝜓)) → (𝜃 → (𝜑𝜓)))
 
Theoremmerlem11 1617 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.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))
 
Theoremmerlem12 1618 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.)
(((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → 𝜑)
 
Theoremmerlem13 1619 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.)
((𝜑𝜓) → (((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → 𝜓))
 
Theoremluk-1 1620 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.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremluk-2 1621 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.)
((¬ 𝜑𝜑) → 𝜑)
 
Theoremluk-3 1622 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.)
(𝜑 → (¬ 𝜑𝜓))
 
1.3.3  Derive the standard axioms from the Lukasiewicz axioms
 
Theoremluklem1 1623 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 23-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)
 
Theoremluklem2 1624 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ¬ 𝜓) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))
 
Theoremluklem3 1625 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (((¬ 𝜑𝜓) → 𝜒) → (𝜃𝜒)))
 
Theoremluklem4 1626 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((((¬ 𝜑𝜑) → 𝜑) → 𝜓) → 𝜓)
 
Theoremluklem5 1627 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
Theoremluklem6 1628 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))
 
Theoremluklem7 1629 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
 
Theoremluklem8 1630 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))
 
Theoremax1 1631 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
Theoremax2 1632 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremax3 1633 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))
 
1.3.4  Derive Nicod's axiom from the standard axioms

Prove Nicod's axiom and implication and negation definitions.

 
Theoremnic-dfim 1634 Define implication in terms of 'nand'. Analogous to ((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓)). 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.)
(((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓))))
 
Theoremnic-dfneg 1635 Define negation in terms of 'nand'. Analogous to ((𝜑𝜑) ↔ ¬ 𝜑). 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.)
(((𝜑𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))
 
Theoremnic-mp 1636 Derive Nicod's rule of modus ponens using 'nand', from the standard one. Although the major and minor premise together also imply 𝜒, this form is necessary for useful derivations from nic-ax 1638. 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.)
𝜑    &   (𝜑 ⊼ (𝜒𝜓))       𝜓
 
Theoremnic-mpALT 1637 A direct proof of nic-mp 1636. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑 ⊼ (𝜒𝜓))       𝜓
 
Theoremnic-ax 1638 Nicod's axiom derived from the standard ones. See Introduction to Mathematical Philosophy by B. Russell, p. 152. Like meredith 1606, the usual axioms can be derived from this and vice versa. Unlike meredith 1606, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g. { nic-ax 1638, nic-mp 1636 } is equivalent to { luk-1 1620, luk-2 1621, luk-3 1622, ax-mp 5 }. 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.)
((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
 
Theoremnic-axALT 1639 A direct proof of nic-ax 1638. (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
 
1.3.5  Derive the Lukasiewicz axioms from Nicod's axiom
 
Theoremnic-imp 1640 Inference for nic-mp 1636 using nic-ax 1638 as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜒𝜓))       ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))
 
Theoremnic-idlem1 1641 Lemma for nic-id 1643. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜃 ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ (((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃) ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃)))
 
Theoremnic-idlem2 1642 Lemma for nic-id 1643. Inference used by nic-id 1643. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜂 ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃))       ((𝜃 ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ 𝜂)
 
Theoremnic-id 1643 Theorem id 22 expressed with . (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜏 ⊼ (𝜏𝜏))
 
Theoremnic-swap 1644 The connector is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜃𝜑) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))
 
Theoremnic-isw1 1645 Inference version of nic-swap 1644. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃𝜑)       (𝜑𝜃)
 
Theoremnic-isw2 1646 Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 ⊼ (𝜃𝜑))       (𝜓 ⊼ (𝜑𝜃))
 
Theoremnic-iimp1 1647 Inference version of nic-imp 1640 using right-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜒𝜓))    &   (𝜃𝜒)       (𝜃𝜑)
 
Theoremnic-iimp2 1648 Inference version of nic-imp 1640 using left-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ⊼ (𝜒𝜒))    &   (𝜃𝜑)       (𝜃 ⊼ (𝜒𝜒))
 
Theoremnic-idel 1649 Inference to remove the trailing term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜒𝜓))       (𝜑 ⊼ (𝜒𝜒))
 
Theoremnic-ich 1650 Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜓𝜓))    &   (𝜓 ⊼ (𝜒𝜒))       (𝜑 ⊼ (𝜒𝜒))
 
Theoremnic-idbl 1651 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.)
(𝜑 ⊼ (𝜓𝜓))       ((𝜓𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜑𝜑)))
 
Theoremnic-bijust 1652 Biconditional justification from Nicod's axiom. For nic-* definitions, the biconditional connective is not used. Instead, definitions are made based on this form. nic-bi1 1653 and nic-bi2 1654 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.)
((𝜏𝜏) ⊼ ((𝜏𝜏) ⊼ (𝜏𝜏)))
 
Theoremnic-bi1 1653 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.)
((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓)))       (𝜑 ⊼ (𝜓𝜓))
 
Theoremnic-bi2 1654 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.)
((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓)))       (𝜓 ⊼ (𝜑𝜑))
 
Theoremnic-stdmp 1655 Derive the standard modus ponens from nic-mp 1636. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓
 
Theoremnic-luk1 1656 Proof of luk-1 1620 from nic-ax 1638 and nic-mp 1636 (and definitions nic-dfim 1634 and nic-dfneg 1635). Note that the standard axioms ax-1 6, ax-2 7, and ax-3 8 are proved from the Lukasiewicz axioms by theorems ax1 1631, ax2 1632, and ax3 1633. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremnic-luk2 1657 Proof of luk-2 1621 from nic-ax 1638 and nic-mp 1636. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) → 𝜑)
 
Theoremnic-luk3 1658 Proof of luk-3 1622 from nic-ax 1638 and nic-mp 1636. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (¬ 𝜑𝜓))
 
1.3.6  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom
 
Theoremlukshef-ax1 1659 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 1636 for its constructions.

Here, the axiom is proved as a substitution instance of nic-ax 1638. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜃 ⊼ (𝜃𝜃)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
 
Theoremlukshefth1 1660 Lemma for renicax 1662. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (𝜑 ⊼ (𝜓𝜒)))
 
Theoremlukshefth2 1661 Lemma for renicax 1662. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜏𝜃) ⊼ ((𝜃𝜏) ⊼ (𝜃𝜏)))
 
Theoremrenicax 1662 A rederivation of nic-ax 1638 from lukshef-ax1 1659, proving that lukshef-ax1 1659 with nic-mp 1636 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.)
((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
 
1.3.7  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms
 
Theoremtbw-bijust 1663 Justification for tbw-negdf 1664. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ (((𝜑𝜓) → ((𝜓𝜑) → ⊥)) → ⊥))
 
Theoremtbw-negdf 1664 The definition of negation, in terms of and . (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)
 
Theoremtbw-ax1 1665 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.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremtbw-ax2 1666 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.)
(𝜑 → (𝜓𝜑))
 
Theoremtbw-ax3 1667 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.)
(((𝜑𝜓) → 𝜑) → 𝜑)
 
Theoremtbw-ax4 1668 The fourth of four axioms in the Tarski-Bernays-Wajsberg system.

This axiom was added to the Tarski-Bernays axiom system (see tb-ax1 32503, tb-ax2 32504, and tb-ax3 32505) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(⊥ → 𝜑)
 
Theoremtbwsyl 1669 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.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)
 
Theoremtbwlem1 1670 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.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
 
Theoremtbwlem2 1671 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.)
((𝜑 → (𝜓 → ⊥)) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))
 
Theoremtbwlem3 1672 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.)
(((((𝜑 → ⊥) → 𝜑) → 𝜑) → 𝜓) → 𝜓)
 
Theoremtbwlem4 1673 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.)
(((𝜑 → ⊥) → 𝜓) → ((𝜓 → ⊥) → 𝜑))
 
Theoremtbwlem5 1674 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.)
(((𝜑 → (𝜓 → ⊥)) → ⊥) → 𝜑)
 
Theoremre1luk1 1675 luk-1 1620 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremre1luk2 1676 luk-2 1621 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) → 𝜑)
 
Theoremre1luk3 1677 luk-3 1622 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1675 and re1luk2 1676 proves that tbw-ax1 1665, tbw-ax2 1666, tbw-ax3 1667, and tbw-ax4 1668, with ax-mp 5 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.)

(𝜑 → (¬ 𝜑𝜓))
 
1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom
 
Theoremmerco1 1678 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 1606 has 21 symbols, sans parentheses.

See merco2 1701 for another axiom of equal length. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(((((𝜑𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏𝜑) → (𝜒𝜑)))
 
Theoremmerco1lem1 1679 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (⊥ → 𝜒))
 
Theoremretbwax4 1680 tbw-ax4 1668 rederived from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊥ → 𝜑)
 
Theoremretbwax2 1681 tbw-ax2 1666 rederived from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
Theoremmerco1lem2 1682 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (((𝜓𝜏) → (𝜑 → ⊥)) → 𝜒))
 
Theoremmerco1lem3 1683 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑))
 
Theoremmerco1lem4 1684 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))
 
Theoremmerco1lem5 1685 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑𝜏))
 
Theoremmerco1lem6 1686 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜒 → (𝜑𝜓)))
 
Theoremmerco1lem7 1687 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (((𝜓𝜒) → 𝜓) → 𝜓))
 
Theoremretbwax3 1688 tbw-ax3 1667 rederived from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)
 
Theoremmerco1lem8 1689 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓 → (𝜓𝜒)) → (𝜓𝜒)))
 
Theoremmerco1lem9 1690 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))
 
Theoremmerco1lem10 1691 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((((𝜑𝜓) → 𝜒) → (𝜏𝜒)) → 𝜑) → (𝜃𝜑))
 
Theoremmerco1lem11 1692 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜒 → (𝜑𝜏)) → ⊥) → 𝜓))
 
Theoremmerco1lem12 1693 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜓))
 
Theoremmerco1lem13 1694 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → (𝜒𝜓)) → 𝜏) → (𝜑𝜏))
 
Theoremmerco1lem14 1695 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → 𝜓) → 𝜒) → (𝜑𝜒))
 
Theoremmerco1lem15 1696 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))
 
Theoremmerco1lem16 1697 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 → (𝜓𝜒)) → 𝜏) → ((𝜑𝜒) → 𝜏))
 
Theoremmerco1lem17 1698 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((((𝜑𝜓) → 𝜑) → 𝜒) → 𝜏) → ((𝜑𝜒) → 𝜏))
 
Theoremmerco1lem18 1699 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜓𝜑) → (𝜓𝜒)))
 
Theoremretbwax1 1700 tbw-ax1 1665 rederived from merco1 1678.

This theorem, along with retbwax2 1681, retbwax3 1688, and retbwax4 1680, shows that merco1 1678 with ax-mp 5 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.)

((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
  Copyright terms: Public domain < Previous  Next >