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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)))

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