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
1.3.5  Derive Nicod's axiom from the standard axioms Prove Nicod's axiom and implication and negation definitions.
Theorem	nic-dfim 1701	This theorem "defines" implication in terms of 'nand'. Analogous to nanim 1527. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓))))
Theorem	nic-dfneg 1702	This theorem "defines" negation in terms of 'nand'. Analogous to nannot 1528. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 ⊼ 𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))
Theorem	nic-mp 1703	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 1705. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))    ⇒   ⊢ 𝜓
Theorem	nic-mpALT 1704	A direct proof of nic-mp 1703. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))    ⇒   ⊢ 𝜓
Theorem	nic-ax 1705	Nicod's axiom derived from the standard ones. See Introduction to Mathematical Philosophy by B. Russell, p. 152. Like meredith 1673, the usual axioms can be derived from this and vice versa. Unlike meredith 1673, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g., { nic-ax 1705, nic-mp 1703 } is equivalent to { luk-1 1687, luk-2 1688, luk-3 1689, ax-mp 5 }. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
Theorem	nic-axALT 1706	A direct proof of nic-ax 1705. (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 1707	Inference for nic-mp 1703 using nic-ax 1705 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 1708	Lemma for nic-id 1710. (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 1709	Lemma for nic-id 1710. Inference used by nic-id 1710. (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 1710	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 1711	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 1712	Inference version of nic-swap 1711. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜃 ⊼ 𝜑)    ⇒   ⊢ (𝜑 ⊼ 𝜃)
Theorem	nic-isw2 1713	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 1714	Inference version of nic-imp 1707 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 1715	Inference version of nic-imp 1707 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 1716	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 1717	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 1718	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 1719	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 1720 and nic-bi2 1721 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 1720	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 1721	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 1722	Derive the standard modus ponens from nic-mp 1703. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ 𝜓
Theorem	nic-luk1 1723	Proof of luk-1 1687 from nic-ax 1705 and nic-mp 1703 (and Definitions nic-dfim 1701 and nic-dfneg 1702). Note that the standard axioms ax-1 6, ax-2 7, and ax-3 8 are proved from the Lukasiewicz axioms by Theorems ax1 1698, ax2 1699, and ax3 1700. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	nic-luk2 1724	Proof of luk-2 1688 from nic-ax 1705 and nic-mp 1703. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	nic-luk3 1725	Proof of luk-3 1689 from nic-ax 1705 and nic-mp 1703. (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 1726	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 1703 for its constructions. Here, the axiom is proved as a substitution instance of nic-ax 1705. (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 1727	Lemma for renicax 1729. (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 1728	Lemma for renicax 1729. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜏 ⊼ 𝜃) ⊼ ((𝜃 ⊼ 𝜏) ⊼ (𝜃 ⊼ 𝜏)))
Theorem	renicax 1729	A rederivation of nic-ax 1705 from lukshef-ax1 1726, proving that lukshef-ax1 1726 with nic-mp 1703 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 1730	Justification for tbw-negdf 1731. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ (((𝜑 → 𝜓) → ((𝜓 → 𝜑) → ⊥)) → ⊥))
Theorem	tbw-negdf 1731	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 1732	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 1733	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 1734	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 1735	The fourth of four axioms in the Tarski-Bernays-Wajsberg system. This axiom was added to the Tarski-Bernays axiom system (see tb-ax1 37263, tb-ax2 37264, and tb-ax3 37265) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊥ → 𝜑)
Theorem	tbwsyl 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)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	tbwlem1 1737	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 1738	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 1739	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 1740	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 1741	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 1742	luk-1 1687 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 1743	luk-2 1688 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 1744	luk-3 1689 derived from the Tarski-Bernays-Wajsberg axioms. This theorem, along with re1luk1 1742 and re1luk2 1743 proves that tbw-ax1 1732, tbw-ax2 1733, tbw-ax3 1734, and tbw-ax4 1735, 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 1745	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 1673 has 21 symbols, sans parentheses. See merco2 1768 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 1746	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (⊥ → 𝜒))
Theorem	retbwax4 1747	tbw-ax4 1735 rederived from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊥ → 𝜑)
Theorem	retbwax2 1748	tbw-ax2 1733 rederived from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	merco1lem2 1749	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (((𝜓 → 𝜏) → (𝜑 → ⊥)) → 𝜒))
Theorem	merco1lem3 1750	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑))
Theorem	merco1lem4 1751	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
Theorem	merco1lem5 1752	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑 → 𝜏))
Theorem	merco1lem6 1753	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜒 → (𝜑 → 𝜓)))
Theorem	merco1lem7 1754	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜓) → 𝜓))
Theorem	retbwax3 1755	tbw-ax3 1734 rederived from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	merco1lem8 1756	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ((𝜓 → (𝜓 → 𝜒)) → (𝜓 → 𝜒)))
Theorem	merco1lem9 1757	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
Theorem	merco1lem10 1758	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (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 1759	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → ⊥) → 𝜓))
Theorem	merco1lem12 1760	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜓))
Theorem	merco1lem13 1761	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → 𝜏) → (𝜑 → 𝜏))
Theorem	merco1lem14 1762	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) → (𝜑 → 𝜒))
Theorem	merco1lem15 1763	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))
Theorem	merco1lem16 1764	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜏) → ((𝜑 → 𝜒) → 𝜏))
Theorem	merco1lem17 1765	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (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 1766	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1745. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜓 → 𝜑) → (𝜓 → 𝜒)))
Theorem	retbwax1 1767	tbw-ax1 1732 rederived from merco1 1745. This theorem, along with retbwax2 1748, retbwax3 1755, and retbwax4 1747, shows that merco1 1745 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 1768	A single axiom for propositional calculus discovered by C. A. Meredith. This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1745. (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 1769	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1768. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))
Theorem	mercolem2 1770	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1768. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))
Theorem	mercolem3 1771	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1768. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))
Theorem	mercolem4 1772	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1768. (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 1773	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1768. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜃 → ((𝜃 → 𝜑) → (𝜏 → (𝜒 → 𝜑))))
Theorem	mercolem6 1774	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1768. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜓 → (𝜑 → 𝜒)))
Theorem	mercolem7 1775	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1768. (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 1776	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1768. (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 1777	tbw-ax1 1732 rederived from merco2 1768. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	re1tbw2 1778	tbw-ax2 1733 rederived from merco2 1768. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	re1tbw3 1779	tbw-ax3 1734 rederived from merco2 1768. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	re1tbw4 1780	tbw-ax4 1735 rederived from merco2 1768. This theorem, along with re1tbw1 1777, re1tbw2 1778, and re1tbw3 1779, shows that merco2 1768, 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 1781	Justification for rb-imdf 1782. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)))
Theorem	rb-imdf 1782	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 1783	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 1784	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 1785	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 1786	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 1787	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 1788	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 1789	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 1790	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 1791	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 1792	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 1793	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 1794	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 1795	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 1796	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 1797	luk-1 1687 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 1798	luk-2 1688 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	re2luk3 1799	luk-3 1689 derived from Russell-Bernays'. This theorem, along with re1axmp 1796, re2luk1 1797, and re2luk2 1798 shows that rb-ax1 1784, rb-ax2 1785, rb-ax3 1786, and rb-ax4 1787, along with anmp 1783, 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 1800, modus ponendo tollens II mptxor 1801, and modus tollendo ponens (exclusive-or version) mtpxor 1803. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1803 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 1802. 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 1804 and stoic1b 1805) and thema 3 (stoic3 1808). 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 1806, stoic2b 1807, stoic4a 1809, and stoic4b 1810. 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 1556. [Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1543 and ancom 466) 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 2687 and related theorems.
Theorem	mptnan 1800	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 1801) 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