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Theorem List for Metamath Proof Explorer - 1701-1800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtbwlem5 1701 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 1702 luk-1 1647 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 1703 luk-2 1648 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 1704 luk-3 1649 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1702 and re1luk2 1703 proves that tbw-ax1 1692, tbw-ax2 1693, tbw-ax3 1694, and tbw-ax4 1695, 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))
 
1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom
 
Theoremmerco1 1705 A single axiom for propositional calculus discovered by C. A. Meredith.

This axiom is worthy of note, due to it having only 19 symbols, not counting parentheses. The more well-known meredith 1633 has 21 symbols, sans parentheses.

See merco2 1728 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 1706 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (⊥ → 𝜒))
 
Theoremretbwax4 1707 tbw-ax4 1695 rederived from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊥ → 𝜑)
 
Theoremretbwax2 1708 tbw-ax2 1693 rederived from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
Theoremmerco1lem2 1709 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (((𝜓𝜏) → (𝜑 → ⊥)) → 𝜒))
 
Theoremmerco1lem3 1710 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑))
 
Theoremmerco1lem4 1711 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))
 
Theoremmerco1lem5 1712 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑𝜏))
 
Theoremmerco1lem6 1713 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜒 → (𝜑𝜓)))
 
Theoremmerco1lem7 1714 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (((𝜓𝜒) → 𝜓) → 𝜓))
 
Theoremretbwax3 1715 tbw-ax3 1694 rederived from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)
 
Theoremmerco1lem8 1716 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓 → (𝜓𝜒)) → (𝜓𝜒)))
 
Theoremmerco1lem9 1717 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))
 
Theoremmerco1lem10 1718 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((((𝜑𝜓) → 𝜒) → (𝜏𝜒)) → 𝜑) → (𝜃𝜑))
 
Theoremmerco1lem11 1719 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜒 → (𝜑𝜏)) → ⊥) → 𝜓))
 
Theoremmerco1lem12 1720 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜓))
 
Theoremmerco1lem13 1721 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → (𝜒𝜓)) → 𝜏) → (𝜑𝜏))
 
Theoremmerco1lem14 1722 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → 𝜓) → 𝜒) → (𝜑𝜒))
 
Theoremmerco1lem15 1723 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))
 
Theoremmerco1lem16 1724 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 → (𝜓𝜒)) → 𝜏) → ((𝜑𝜒) → 𝜏))
 
Theoremmerco1lem17 1725 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((((𝜑𝜓) → 𝜑) → 𝜒) → 𝜏) → ((𝜑𝜒) → 𝜏))
 
Theoremmerco1lem18 1726 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1705. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜓𝜑) → (𝜓𝜒)))
 
Theoremretbwax1 1727 tbw-ax1 1692 rederived from merco1 1705.

This theorem, along with retbwax2 1708, retbwax3 1715, and retbwax4 1707, shows that merco1 1705 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)))
 
1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom
 
Theoremmerco2 1728 A single axiom for propositional calculus discovered by C. A. Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1705. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(((𝜑𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃𝜑) → (𝜏 → (𝜂𝜑))))
 
Theoremmercolem1 1729 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒)))
 
Theoremmercolem2 1730 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → (𝜒 → (𝜃𝜑)))
 
Theoremmercolem3 1731 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜒) → (𝜓 → (𝜑𝜒)))
 
Theoremmercolem4 1732 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))
 
Theoremmercolem5 1733 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 → ((𝜃𝜑) → (𝜏 → (𝜒𝜑))))
 
Theoremmercolem6 1734 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))
 
Theoremmercolem7 1735 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓)))
 
Theoremmercolem8 1736 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))
 
Theoremre1tbw1 1737 tbw-ax1 1692 rederived from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremre1tbw2 1738 tbw-ax2 1693 rederived from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
Theoremre1tbw3 1739 tbw-ax3 1694 rederived from merco2 1728. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)
 
Theoremre1tbw4 1740 tbw-ax4 1695 rederived from merco2 1728.

This theorem, along with re1tbw1 1737, re1tbw2 1738, and re1tbw3 1739, shows that merco2 1728, along with ax-mp 5, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(⊥ → 𝜑)
 
1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms
 
Theoremrb-bijust 1741 Justification for rb-imdf 1742. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)))
 
Theoremrb-imdf 1742 The definition of implication, in terms of and ¬. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (¬ (¬ (𝜑𝜓) ∨ (¬ 𝜑𝜓)) ∨ ¬ (¬ (¬ 𝜑𝜓) ∨ (𝜑𝜓)))
 
Theoremanmp 1743 Modus ponens for ¬ axiom systems. (Contributed by Anthony Hart, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜑𝜓)       𝜓
 
Theoremrb-ax1 1744 The first of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (¬ 𝜓𝜒) ∨ (¬ (𝜑𝜓) ∨ (𝜑𝜒)))
 
Theoremrb-ax2 1745 The second of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (𝜑𝜓) ∨ (𝜓𝜑))
 
Theoremrb-ax3 1746 The third of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑 ∨ (𝜓𝜑))
 
Theoremrb-ax4 1747 The fourth of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (𝜑𝜑) ∨ 𝜑)
 
Theoremrbsyl 1748 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜓𝜒)    &   (𝜑𝜓)       (𝜑𝜒)
 
Theoremrblem1 1749 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑𝜓)    &   𝜒𝜃)       (¬ (𝜑𝜒) ∨ (𝜓𝜃))
 
Theoremrblem2 1750 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (𝜒𝜑) ∨ (𝜒 ∨ (𝜑𝜓)))
 
Theoremrblem3 1751 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (𝜒𝜑) ∨ ((𝜒𝜓) ∨ 𝜑))
 
Theoremrblem4 1752 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑𝜃)    &   𝜓𝜏)    &   𝜒𝜂)       (¬ ((𝜑𝜓) ∨ 𝜒) ∨ ((𝜂𝜏) ∨ 𝜃))
 
Theoremrblem5 1753 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (¬ ¬ 𝜑𝜓) ∨ (¬ ¬ 𝜓𝜑))
 
Theoremrblem6 1754 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))       𝜑𝜓)
 
Theoremrblem7 1755 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))       𝜓𝜑)
 
Theoremre1axmp 1756 ax-mp 5 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓
 
Theoremre2luk1 1757 luk-1 1647 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremre2luk2 1758 luk-2 1648 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) → 𝜑)
 
Theoremre2luk3 1759 luk-3 1649 derived from Russell-Bernays'.

This theorem, along with re1axmp 1756, re2luk1 1757, and re2luk2 1758 shows that rb-ax1 1744, rb-ax2 1745, rb-ax3 1746, and rb-ax4 1747, along with anmp 1743, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (¬ 𝜑𝜓))
 
1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)

The Greek Stoics developed a system of logic called Stoic logic. The Stoic Chrysippus, in particular, was often considered one of the greatest logicians of antiquity. Stoic logic is different from Aristotle's system, since it focuses on propositional logic, though later thinkers did combine the systems of the Stoics with Aristotle. Jan Lukasiewicz reports, "For anybody familiar with mathematical logic it is self-evident that the Stoic dialectic is the ancient form of modern propositional logic" ( On the history of the logic of proposition by Jan Lukasiewicz (1934), translated in: Selected Works - Edited by Ludwik Borkowski - Amsterdam, North-Holland, 1970 pp. 197-217, referenced in "History of Logic, https://www.historyoflogic.com/logic-stoics.htm).

In this section we show that the propositional logic system we use (which is non-modal) is at least as strong as the non-modal portion of Stoic logic. We show this by showing that our system assumes or proves all of key features of Stoic logic's non-modal portion (specifically the Stoic logic indemonstrables, themata, and principles).

"In terms of contemporary logic, Stoic syllogistic is best understood as a substructural backwards-working Gentzen-style natural-deduction system that consists of five kinds of axiomatic arguments (the indemonstrables) and four inference rules, called themata. An argument is a syllogism precisely if it either is an indemonstrable or can be reduced to one by means of the themata (Diogenes Laertius (D. L. 7.78))." (Ancient Logic, Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/logic-ancient/). There are also a few "principles" that support logical reasoning, discussed below. For more information, see "Stoic Logic" by Susanne Bobzien, especially [Bobzien] p. 110-120, especially for a discussion about the themata (including how they were reconstructed and how they were used). There are differences in the systems we can only partly represent, for example, in Stoic logic "truth and falsehood are temporal properties of assertibles... They can belong to an assertible at one time but not at another" ([Bobzien] p. 87). Stoic logic also included various kinds of modalities, which we do not include here since our basic propositional logic does not include modalities.

A key part of the Stoic logic system is a set of five "indemonstrables" assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is difficult to assign specific ideas to specific thinkers. The indemonstrables are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus ponens) ax-mp 5, modus tollendo tollens (modus tollens) mto 198, modus ponendo tollens I mptnan 1760, modus ponendo tollens II mptxor 1761, and modus tollendo ponens (exclusive-or version) mtpxor 1763. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1763 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtpor 1762.

After we prove the indemonstratables, we then prove all the Stoic logic themata (the inference rules of Stoic logic; "thema" is singular). This is straightforward for thema 1 (stoic1a 1764 and stoic1b 1765) and thema 3 (stoic3 1768). However, while Stoic logic was once a leading logic system, most direct information about Stoic logic has since been lost, including the exact texts of thema 2 and thema 4. There are, however, enough references and specific examples to support reconstruction. Themata 2 and 4 have been reconstructed; see statements T2 and T4 in [Bobzien] p. 110-120 and our proofs of them in stoic2a 1766, stoic2b 1767, stoic4a 1769, and stoic4b 1770.

Stoic logic also had a set of principles involving assertibles. Statements in [Bobzien] p. 99 express the known principles. The following paragraphs discuss these principles and our proofs of them.

"A principle of double negation, expressed by saying that a double-negation (Not: not: p) is equivalent to the assertible that is doubly negated (p) (DL VII 69)." In other words, (𝜑 ↔ ¬ ¬ 𝜑) as proven in notnotb 316.

"The principle that all conditionals that are formed by using the same assertible twice (like 'If p, p') are true (Cic. Acad. II 98)." In other words, (𝜑𝜑) as proven in id 22.

"The principle that all disjunctions formed by a contradiction (like 'Either p or not: p') are true (S. E. M VIII 282)." Remember that in Stoic logic, 'or' means 'exclusive or'. In other words, (𝜑 ⊻ ¬ 𝜑) as proven in xorexmid 1511.

[Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1498 and ancom 461) and the principle of contraposition (con4 113) (pointing to DL VII 194).

In short, the non-modal propositional logic system we use is at least as strong as the non-modal portion of Stoic logic.

For more about Aristotle's system, see barbara 2746 and related theorems.

 
Theoremmptnan 1760 Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mptxor 1761) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.)
𝜑    &    ¬ (𝜑𝜓)        ¬ 𝜓
 
Theoremmptxor 1761 Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or . See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 12-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.)
𝜑    &   (𝜑𝜓)        ¬ 𝜓
 
Theoremmtpor 1762 Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1763, one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if 𝜑 is not true, and 𝜑 or 𝜓 (or both) are true, then 𝜓 must be true". An alternate phrasing is: "once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth". -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.)
¬ 𝜑    &   (𝜑𝜓)       𝜓
 
Theoremmtpxor 1763 Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1762, one of the five "indemonstrables" in Stoic logic. The rule says: "if 𝜑 is not true, and either 𝜑 or 𝜓 (exclusively) are true, then 𝜓 must be true". Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1762. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1761, that is, it is exclusive-or df-xor 1496), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mptxor 1761), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.)
¬ 𝜑    &   (𝜑𝜓)       𝜓
 
Theoremstoic1a 1764 Stoic logic Thema 1 (part a).

The first thema of the four Stoic logic themata, in its basic form, was:

"When from two (assertibles) a third follows, then from either of them together with the contradictory of the conclusion the contradictory of the other follows." (Apuleius Int. 209.9-14), see [Bobzien] p. 117 and https://plato.stanford.edu/entries/logic-ancient/

We will represent thema 1 as two very similar rules stoic1a 1764 and stoic1b 1765 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.)

((𝜑𝜓) → 𝜃)       ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)
 
Theoremstoic1b 1765 Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1764. (Contributed by David A. Wheeler, 16-Feb-2019.)
((𝜑𝜓) → 𝜃)       ((𝜓 ∧ ¬ 𝜃) → ¬ 𝜑)
 
Theoremstoic2a 1766 Stoic logic Thema 2 version a. Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 2 as follows: "When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two." Bobzien uses constructs such as 𝜑, 𝜓𝜒; in Metamath we will represent that construct as 𝜑𝜓𝜒. This version a is without the phrase "or both"; see stoic2b 1767 for the version with the phrase "or both". We already have this rule as syldan 591, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremstoic2b 1767 Stoic logic Thema 2 version b. See stoic2a 1766. Version b is with the phrase "or both". We already have this rule as mpd3an3 1453, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremstoic3 1768 Stoic logic Thema 3. Statement T3 of [Bobzien] p. 116-117 discusses Stoic logic Thema 3. "When from two (assemblies) a third follows, and from the one that follows (i.e., the third) together with another, external assumption, another follows, then that other follows from the first two and the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   ((𝜒𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)
 
Theoremstoic4a 1769 Stoic logic Thema 4 version a. Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic Thema 4: "When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s)."

We use 𝜃 to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1770 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)

((𝜑𝜓) → 𝜒)    &   ((𝜒𝜑𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)
 
Theoremstoic4b 1770 Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1769 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   (((𝜒𝜑𝜓) ∧ 𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)
 
1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)

Here we extend the language of wffs with predicate calculus, which allows us to talk about individual objects in a domain of discourse (which for us will be the universe of all sets, so we call them "setvar variables") and make true/false statements about predicates, which are relationships between objects, such as whether or not two objects are equal. In addition, we introduce universal quantification ("for all", e.g. ax-4 1801) in order to make statements about whether a wff holds for every object in the domain of discourse. Later we introduce existential quantification ("there exists", df-ex 1772) which is defined in terms of universal quantification.

Our axioms are really axiom schemes, and our wff and setvar variables are metavariables ranging over expressions in an underlying "object language". This is explained here: mmset.html#axiomnote 1772.

Our axiom system starts with the predicate calculus axiom schemes system S2 of Tarski defined in his 1965 paper, "A Simplified Formalization of Predicate Logic with Identity" [Tarski]. System S2 is defined in the last paragraph on p. 77, and repeated on p. 81 of [KalishMontague]. We do not include scheme B5 (our sp 2172) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 2032 below).

Theorem spw 2032 can be used to prove any instance of sp 2172 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 2172 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 2172 as part of what we call "Tarski's system" because we want it to be the smallest set of axioms that is logically complete with no redundancies. We later prove sp 2172 as theorem axc5 35911 using the auxiliary axiom schemes that make our system metalogically complete.

Our version of Tarski's system S2 consists of propositional calculus (ax-mp 5, ax-1 6, ax-2 7, ax-3 8) plus ax-gen 1787, ax-4 1801, ax-5 1902, ax-6 1961, ax-7 2006, ax-8 2107, and ax-9 2115. The last three are equality axioms that represent three sub-schemes of Tarski's scheme B8. Due to its side-condition ("where 𝜑 is an atomic formula and 𝜓 is obtained by replacing an occurrence of the variable 𝑥 by the variable 𝑦"), we cannot represent his B8 directly without greatly complicating our scheme language, but the simpler schemes ax-7 2006, ax-8 2107, and ax-9 2115 are sufficient for set theory and much easier to work with.

Tarski's system is exactly equivalent to the traditional axiom system in most logic textbooks but has the advantage of being easy to manipulate with a computer program, and its simpler metalogic (with no built-in notions of "free variable" and "proper substitution") is arguably easier for a non-logician human to follow step by step in a proof (where "follow" means being able to identify the substitutions that were made, without necessarily a higher-level understanding). In particular, it is logically complete in that it can derive all possible object-language theorems of predicate calculus with equality, i.e., the same theorems as the traditional system can derive.

However, for efficiency (and indeed a key feature that makes Metamath successful), our system is designed to derive reusable theorem schemes (rather than object-language theorems) from other schemes. From this "metalogical" point of view, Tarski's S2 is not complete. For example, we cannot derive scheme sp 2172, even though (using spw 2032) we can derive all instances of it that do not involve wff metavariables or bundled setvar variables. (Two setvar variables are "bundled" if they can be substituted with the same setvar variable, i.e., do not have a "$d" disjoint variable condition.) Later we will introduce auxiliary axiom schemes ax-10 2136, ax-11 2151, ax-12 2167, and ax-13 2383 that are metatheorems of Tarski's system (i.e. are logically redundant) but which give our system the property of "scheme completeness", allowing us to prove directly (instead of, say, by induction on formula length) all possible schemes that can be expressed in our language.

 
1.4.1  Universal quantifier (continued); define "exists" and "not free"

The universal quantifier was introduced above in wal 1526 for use by df-tru 1531. See the comments in that section. In this section, we continue with the first "real" use of it.

 
1.4.1.1  Existential quantifier
 
Syntaxwex 1771 Extend wff definition to include the existential quantifier ("there exists").
wff 𝑥𝜑
 
Definitiondf-ex 1772 Define existential quantification. 𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true". Dual of alex 1817. See also the dual pair alnex 1773 / exnal 1818. Definition of [Margaris] p. 49. (Contributed by NM, 10-Jan-1993.)
(∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
 
Theoremalnex 1773 Universal quantification of negation is equivalent to negation of existential quantification. Dual of exnal 1818 (but does not depend on ax-4 1801 contrary to it). See also the dual pair df-ex 1772 / alex 1817. Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
(∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
 
Theoremeximal 1774 An equivalence between an implication with an existentially quantified antecedent and an implication with a universally quantified consequent. An interesting case is when the same formula is substituted for both 𝜑 and 𝜓, since then both implications express a type of non-freeness. See also alimex 1822. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
 
1.4.1.2  Non-freeness predicate
 
Syntaxwnf 1775 Extend wff definition to include the not-free predicate.
wff 𝑥𝜑
 
Definitiondf-nf 1776 Define the not-free predicate for wffs. This is read "𝑥 is not free in 𝜑". Not-free means that the value of 𝑥 cannot affect the value of 𝜑, e.g., any occurrence of 𝑥 in 𝜑 is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 2262). An example of where this is used is stdpc5 2199. See nf5 2282 for an alternate definition which involves nested quantifiers on the same variable.

Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition.

To be precise, our definition really means "effectively not free", because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, 𝑥 is effectively not free in the formula 𝑥 = 𝑥 (see nfequid 2011), even though 𝑥 would be considered free in the usual textbook definition, because the value of 𝑥 in the formula 𝑥 = 𝑥 cannot affect the truth of that formula (and thus substitutions will not change the result).

This definition of not-free tightly ties to the quantifier 𝑥. At this state (no axioms restricting quantifiers yet) 'non-free' appears quite arbitrary. Its intended semantics expresses single-valuedness (constness) across a parameter, but is only evolved as much as later axioms assign properties to quantifiers. It seems the definition here is best suited in situations, where axioms are only partially in effect. In particular, this definition more easily carries over to other logic models with weaker axiomization.

The reverse implication of the definiens (the right hand side of the biconditional) always holds, see 19.2 1972.

This predicate only applies to wffs. See df-nfc 2963 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 24-Sep-2016.) Convert to definition. (Revised by BJ, 6-May-2019.)

(Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
 
Theoremnf2 1777 Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
 
Theoremnf3 1778 Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
 
Theoremnf4 1779 Alternate definition of non-freeness. This definition uses only primitive symbols (→ , ¬ , ∀). (Contributed by BJ, 16-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
 
Theoremnfi 1780 Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Wolf Lammen, 15-Sep-2021.)
(∃𝑥𝜑 → ∀𝑥𝜑)       𝑥𝜑
 
Theoremnfri 1781 Consequence of the definition of not-free. (Contributed by Wolf Lammen, 16-Sep-2021.)
𝑥𝜑       (∃𝑥𝜑 → ∀𝑥𝜑)
 
Theoremnfd 1782 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)
(𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
Theoremnfrd 1783 Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
 
Theoremnftht 1784 Closed form of nfth 1793. (Contributed by Wolf Lammen, 19-Aug-2018.) (Proof shortened by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 3-Sep-2022.)
(∀𝑥𝜑 → Ⅎ𝑥𝜑)
 
Theoremnfntht 1785 Closed form of nfnth 1794. (Contributed by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 4-Sep-2022.)
(¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)
 
Theoremnfntht2 1786 Closed form of nfnth 1794. (Contributed by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 4-Sep-2022.)
(∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)
 
1.4.2  Rule scheme ax-gen (Generalization)
 
Axiomax-gen 1787 Rule of (universal) generalization. In our axiomatization, this is the only postulated (that is, axiomatic) rule of inference of predicate calculus (together with the rule of modus ponens ax-mp 5 of propositional calculus). See, e.g., Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved 𝑥 = 𝑥, then we can conclude 𝑥𝑥 = 𝑥 or even 𝑦𝑥 = 𝑥. Theorem altru 1799 shows the special case 𝑥. The converse rule of inference spi 2173 (universal instantiation, or universal specialization) shows that we can also go the other way: in other words, we can add or remove universal quantifiers from the beginning of any theorem as required. Note that the closed form (𝜑 → ∀𝑥𝜑) need not hold (but may hold in special cases, see ax-5 1902). (Contributed by NM, 3-Jan-1993.)
𝜑       𝑥𝜑
 
Theoremgen2 1788 Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
𝜑       𝑥𝑦𝜑
 
Theoremmpg 1789 Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
(∀𝑥𝜑𝜓)    &   𝜑       𝜓
 
Theoremmpgbi 1790 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
(∀𝑥𝜑𝜓)    &   𝜑       𝜓
 
Theoremmpgbir 1791 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
(𝜑 ↔ ∀𝑥𝜓)    &   𝜓       𝜑
 
Theoremnex 1792 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
¬ 𝜑        ¬ ∃𝑥𝜑
 
Theoremnfth 1793 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
𝜑       𝑥𝜑
 
Theoremnfnth 1794 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
¬ 𝜑       𝑥𝜑
 
Theoremhbth 1795 No variable is (effectively) free in a theorem.

This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form (𝜑 → ∀𝑥𝜑) from smaller formulas of this form. These are useful for constructing hypotheses that state "𝑥 is (effectively) not free in 𝜑". (Contributed by NM, 11-May-1993.) This hb* idiom is generally being replaced by the nf* idiom (see nfth 1793), but keeps its interest in some cases. (Revised by BJ, 23-Sep-2022.)

𝜑       (𝜑 → ∀𝑥𝜑)
 
Theoremnftru 1796 The true constant has no free variables. (This can also be proven in one step with nfv 1906, but this proof does not use ax-5 1902.) (Contributed by Mario Carneiro, 6-Oct-2016.)
𝑥
 
Theoremnffal 1797 The false constant has no free variables (see nftru 1796). (Contributed by BJ, 6-May-2019.)
𝑥
 
Theoremsptruw 1798 Version of sp 2172 when 𝜑 is true. Instance of a1i 11. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.)
𝜑       (∀𝑥𝜑𝜑)
 
Theoremaltru 1799 For all sets, is true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥
 
Theoremalfal 1800 For all sets, ¬ ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥 ¬ ⊥
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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 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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