P. 18 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1701-1800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nic-axALT 1701	A direct proof of nic-ax 1700. (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom
Theorem	nic-imp 1702	Inference for nic-mp 1698 using nic-ax 1700 as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))    ⇒   ⊢ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))
Theorem	nic-idlem1 1703	Lemma for nic-id 1705. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜃 ⊼ (𝜏 ⊼ (𝜏 ⊼ 𝜏))) ⊼ (((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ 𝜃) ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ 𝜃)))
Theorem	nic-idlem2 1704	Lemma for nic-id 1705. Inference used by nic-id 1705. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜂 ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ 𝜃))    ⇒   ⊢ ((𝜃 ⊼ (𝜏 ⊼ (𝜏 ⊼ 𝜏))) ⊼ 𝜂)
Theorem	nic-id 1705	Theorem id 23 expressed with ⊼. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜏 ⊼ (𝜏 ⊼ 𝜏))
Theorem	nic-swap 1706	The connector ⊼ is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜃 ⊼ 𝜑) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))
Theorem	nic-isw1 1707	Inference version of nic-swap 1706. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜃 ⊼ 𝜑)    ⇒   ⊢ (𝜑 ⊼ 𝜃)
Theorem	nic-isw2 1708	Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜓 ⊼ (𝜃 ⊼ 𝜑))    ⇒   ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜃))
Theorem	nic-iimp1 1709	Inference version of nic-imp 1702 using right-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))    &   ⊢ (𝜃 ⊼ 𝜒)    ⇒   ⊢ (𝜃 ⊼ 𝜑)
Theorem	nic-iimp2 1710	Inference version of nic-imp 1702 using left-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ⊼ 𝜓) ⊼ (𝜒 ⊼ 𝜒))    &   ⊢ (𝜃 ⊼ 𝜑)    ⇒   ⊢ (𝜃 ⊼ (𝜒 ⊼ 𝜒))
Theorem	nic-idel 1711	Inference to remove the trailing term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))    ⇒   ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜒))
Theorem	nic-ich 1712	Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ⊼ (𝜓 ⊼ 𝜓))    &   ⊢ (𝜓 ⊼ (𝜒 ⊼ 𝜒))    ⇒   ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜒))
Theorem	nic-idbl 1713	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 ⊼ (𝜓 ⊼ 𝜓))    ⇒   ⊢ ((𝜓 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)))
Theorem	nic-bijust 1714	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 1715 and nic-bi2 1716 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.)
⊢ ((𝜏 ⊼ 𝜏) ⊼ ((𝜏 ⊼ 𝜏) ⊼ (𝜏 ⊼ 𝜏)))
Theorem	nic-bi1 1715	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.)
⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)))    ⇒   ⊢ (𝜑 ⊼ (𝜓 ⊼ 𝜓))
Theorem	nic-bi2 1716	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.)
⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)))    ⇒   ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜑))
Theorem	nic-stdmp 1717	Derive the standard modus ponens from nic-mp 1698. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ 𝜓
Theorem	nic-luk1 1718	Proof of luk-1 1682 from nic-ax 1700 and nic-mp 1698 (and Definitions nic-dfim 1696 and nic-dfneg 1697). Note that the standard axioms ax-1 6, ax-2 7, and ax-3 8 are proved from the Lukasiewicz axioms by Theorems ax1 1693, ax2 1694, and ax3 1695. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	nic-luk2 1719	Proof of luk-2 1683 from nic-ax 1700 and nic-mp 1698. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	nic-luk3 1720	Proof of luk-3 1684 from nic-ax 1700 and nic-mp 1698. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (¬ 𝜑 → 𝜓))
1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom
Theorem	lukshef-ax1 1721	This alternative axiom for propositional calculus using the Sheffer Stroke was discovered 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 1698 for its constructions. Here, the axiom is proved as a substitution instance of nic-ax 1700. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜃 ⊼ (𝜃 ⊼ 𝜃)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
Theorem	lukshefth1 1722	Lemma for renicax 1724. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜏 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜏) ⊼ (𝜑 ⊼ 𝜏))) ⊼ (𝜃 ⊼ (𝜃 ⊼ 𝜃))) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜒)))
Theorem	lukshefth2 1723	Lemma for renicax 1724. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜏 ⊼ 𝜃) ⊼ ((𝜃 ⊼ 𝜏) ⊼ (𝜃 ⊼ 𝜏)))
Theorem	renicax 1724	A rederivation of nic-ax 1700 from lukshef-ax1 1721, proving that lukshef-ax1 1721 with nic-mp 1698 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 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms
Theorem	tbw-bijust 1725	Justification for tbw-negdf 1726. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ (((𝜑 → 𝜓) → ((𝜓 → 𝜑) → ⊥)) → ⊥))
Theorem	tbw-negdf 1726	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) → ⊥)) → ⊥)
Theorem	tbw-ax1 1727	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.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	tbw-ax2 1728	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 → (𝜓 → 𝜑))
Theorem	tbw-ax3 1729	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.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	tbw-ax4 1730	The fourth of four axioms in the Tarski-Bernays-Wajsberg system. This axiom was added to the Tarski-Bernays axiom system (see tb-ax1 36783, tb-ax2 36784, and tb-ax3 36785) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊥ → 𝜑)
Theorem	tbwsyl 1731	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 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	tbwlem1 1732	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 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
Theorem	tbwlem2 1733	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 → 𝜒) → 𝜃) → (𝜓 → 𝜃)))
Theorem	tbwlem3 1734	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)
Theorem	tbwlem4 1735	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))
Theorem	tbwlem5 1736	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)
Theorem	re1luk1 1737	luk-1 1682 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	re1luk2 1738	luk-2 1683 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	re1luk3 1739	luk-3 1684 derived from the Tarski-Bernays-Wajsberg axioms. This theorem, along with re1luk1 1737 and re1luk2 1738 proves that tbw-ax1 1727, tbw-ax2 1728, tbw-ax3 1729, and tbw-ax4 1730, 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 → (¬ 𝜑 → 𝜓))
1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom
Theorem	merco1 1740	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 1668 has 21 symbols, sans parentheses. See merco2 1763 for another axiom of equal length. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((((𝜑 → 𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏 → 𝜑) → (𝜒 → 𝜑)))
Theorem	merco1lem1 1741	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (⊥ → 𝜒))
Theorem	retbwax4 1742	tbw-ax4 1730 rederived from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊥ → 𝜑)
Theorem	retbwax2 1743	tbw-ax2 1728 rederived from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	merco1lem2 1744	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (((𝜓 → 𝜏) → (𝜑 → ⊥)) → 𝜒))
Theorem	merco1lem3 1745	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑))
Theorem	merco1lem4 1746	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
Theorem	merco1lem5 1747	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑 → 𝜏))
Theorem	merco1lem6 1748	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜒 → (𝜑 → 𝜓)))
Theorem	merco1lem7 1749	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜓) → 𝜓))
Theorem	retbwax3 1750	tbw-ax3 1729 rederived from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	merco1lem8 1751	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ((𝜓 → (𝜓 → 𝜒)) → (𝜓 → 𝜒)))
Theorem	merco1lem9 1752	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
Theorem	merco1lem10 1753	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((((𝜑 → 𝜓) → 𝜒) → (𝜏 → 𝜒)) → 𝜑) → (𝜃 → 𝜑))
Theorem	merco1lem11 1754	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → ⊥) → 𝜓))
Theorem	merco1lem12 1755	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜓))
Theorem	merco1lem13 1756	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → 𝜏) → (𝜑 → 𝜏))
Theorem	merco1lem14 1757	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) → (𝜑 → 𝜒))
Theorem	merco1lem15 1758	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))
Theorem	merco1lem16 1759	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜏) → ((𝜑 → 𝜒) → 𝜏))
Theorem	merco1lem17 1760	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((((𝜑 → 𝜓) → 𝜑) → 𝜒) → 𝜏) → ((𝜑 → 𝜒) → 𝜏))
Theorem	merco1lem18 1761	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1740. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜓 → 𝜑) → (𝜓 → 𝜒)))
Theorem	retbwax1 1762	tbw-ax1 1727 rederived from merco1 1740. This theorem, along with retbwax2 1743, retbwax3 1750, and retbwax4 1742, shows that merco1 1740 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.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom
Theorem	merco2 1763	A single axiom for propositional calculus discovered by C. A. Meredith. This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1740. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → (𝜏 → (𝜂 → 𝜑))))
Theorem	mercolem1 1764	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))
Theorem	mercolem2 1765	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))
Theorem	mercolem3 1766	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))
Theorem	mercolem4 1767	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜃 → (𝜂 → 𝜑)) → (((𝜃 → 𝜒) → 𝜑) → (𝜏 → (𝜂 → 𝜑))))
Theorem	mercolem5 1768	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜃 → ((𝜃 → 𝜑) → (𝜏 → (𝜒 → 𝜑))))
Theorem	mercolem6 1769	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜓 → (𝜑 → 𝜒)))
Theorem	mercolem7 1770	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓)))
Theorem	mercolem8 1771	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒)))))
Theorem	re1tbw1 1772	tbw-ax1 1727 rederived from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	re1tbw2 1773	tbw-ax2 1728 rederived from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	re1tbw3 1774	tbw-ax3 1729 rederived from merco2 1763. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	re1tbw4 1775	tbw-ax4 1730 rederived from merco2 1763. This theorem, along with re1tbw1 1772, re1tbw2 1773, and re1tbw3 1774, shows that merco2 1763, 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
Theorem	rb-bijust 1776	Justification for rb-imdf 1777. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)))
Theorem	rb-imdf 1777	The definition of implication, in terms of ∨ and ¬. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ (¬ (¬ (𝜑 → 𝜓) ∨ (¬ 𝜑 ∨ 𝜓)) ∨ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜑 → 𝜓)))
Theorem	anmp 1778	Modus ponens for { ∨ , ¬ } axiom systems. (Contributed by Anthony Hart, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (¬ 𝜑 ∨ 𝜓)    ⇒   ⊢ 𝜓
Theorem	rb-ax1 1779	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.)
⊢ (¬ (¬ 𝜓 ∨ 𝜒) ∨ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒)))
Theorem	rb-ax2 1780	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.)
⊢ (¬ (𝜑 ∨ 𝜓) ∨ (𝜓 ∨ 𝜑))
Theorem	rb-ax3 1781	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 ∨ (𝜓 ∨ 𝜑))
Theorem	rb-ax4 1782	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.)
⊢ (¬ (𝜑 ∨ 𝜑) ∨ 𝜑)
Theorem	rbsyl 1783	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝜓 ∨ 𝜒)    &   ⊢ (𝜑 ∨ 𝜓)    ⇒   ⊢ (𝜑 ∨ 𝜒)
Theorem	rblem1 1784	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝜑 ∨ 𝜓)    &   ⊢ (¬ 𝜒 ∨ 𝜃)    ⇒   ⊢ (¬ (𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃))
Theorem	rblem2 1785	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ (𝜒 ∨ 𝜑) ∨ (𝜒 ∨ (𝜑 ∨ 𝜓)))
Theorem	rblem3 1786	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ (𝜒 ∨ 𝜑) ∨ ((𝜒 ∨ 𝜓) ∨ 𝜑))
Theorem	rblem4 1787	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝜑 ∨ 𝜃)    &   ⊢ (¬ 𝜓 ∨ 𝜏)    &   ⊢ (¬ 𝜒 ∨ 𝜂)    ⇒   ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ((𝜂 ∨ 𝜏) ∨ 𝜃))
Theorem	rblem5 1788	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ (¬ ¬ 𝜑 ∨ 𝜓) ∨ (¬ ¬ 𝜓 ∨ 𝜑))
Theorem	rblem6 1789	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑))    ⇒   ⊢ (¬ 𝜑 ∨ 𝜓)
Theorem	rblem7 1790	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑))    ⇒   ⊢ (¬ 𝜓 ∨ 𝜑)
Theorem	re1axmp 1791	ax-mp 5 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ 𝜓
Theorem	re2luk1 1792	luk-1 1682 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	re2luk2 1793	luk-2 1683 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	re2luk3 1794	luk-3 1684 derived from Russell-Bernays'. This theorem, along with re1axmp 1791, re2luk1 1792, and re2luk2 1793 shows that rb-ax1 1779, rb-ax2 1780, rb-ax3 1781, and rb-ax4 1782, along with anmp 1778, 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 → (¬ 𝜑 → 𝜓))
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 200, modus ponendo tollens I mptnan 1795, modus ponendo tollens II mptxor 1796, and modus tollendo ponens (exclusive-or version) mtpxor 1798. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1798 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 1797. 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 1799 and stoic1b 1800) and thema 3 (stoic3 1803). 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 1801, stoic2b 1802, stoic4a 1804, and stoic4b 1805. 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 318. "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, (𝜑 → 𝜑) as proven in id 23. "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 1554. [Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1541 and ancom 465) and the principle of contraposition (con4 114) (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 2696 and related theorems.
Theorem	mptnan 1795	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 1796) 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    &   ⊢ ¬ (𝜑 ∧ 𝜓)    ⇒   ⊢ ¬ 𝜓
Theorem	mptxor 1796	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    &   ⊢ (𝜑 ⊻ 𝜓)    ⇒   ⊢ ¬ 𝜓
Theorem	mtpor 1797	Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1798, 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    &   ⊢ (𝜑 ∨ 𝜓)    ⇒   ⊢ 𝜓
Theorem	mtpxor 1798	Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1797, 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 1797. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1796, that is, it is exclusive-or df-xor 1539), 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 1796), 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    &   ⊢ (𝜑 ⊻ 𝜓)    ⇒   ⊢ 𝜓
Theorem	stoic1a 1799	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 1799 and stoic1b 1800 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)
Theorem	stoic1b 1800	Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1799. (Contributed by David A. Wheeler, 16-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ ¬ 𝜃) → ¬ 𝜑)