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-iimp1 1701	Inference version of nic-imp 1694 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 1702	Inference version of nic-imp 1694 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 1703	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 1704	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 1705	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 1706	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 1707 and nic-bi2 1708 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 1707	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 1708	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 1709	Derive the standard modus ponens from nic-mp 1690. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ 𝜓
Theorem	nic-luk1 1710	Proof of luk-1 1674 from nic-ax 1692 and nic-mp 1690 (and Definitions nic-dfim 1688 and nic-dfneg 1689). Note that the standard axioms ax-1 6, ax-2 7, and ax-3 8 are proved from the Lukasiewicz axioms by Theorems ax1 1685, ax2 1686, and ax3 1687. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	nic-luk2 1711	Proof of luk-2 1675 from nic-ax 1692 and nic-mp 1690. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	nic-luk3 1712	Proof of luk-3 1676 from nic-ax 1692 and nic-mp 1690. (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 1713	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 1690 for its constructions. Here, the axiom is proved as a substitution instance of nic-ax 1692. (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 1714	Lemma for renicax 1716. (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 1715	Lemma for renicax 1716. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜏 ⊼ 𝜃) ⊼ ((𝜃 ⊼ 𝜏) ⊼ (𝜃 ⊼ 𝜏)))
Theorem	renicax 1716	A rederivation of nic-ax 1692 from lukshef-ax1 1713, proving that lukshef-ax1 1713 with nic-mp 1690 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 1717	Justification for tbw-negdf 1718. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ (((𝜑 → 𝜓) → ((𝜓 → 𝜑) → ⊥)) → ⊥))
Theorem	tbw-negdf 1718	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 1719	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 1720	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 1721	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 1722	The fourth of four axioms in the Tarski-Bernays-Wajsberg system. This axiom was added to the Tarski-Bernays axiom system (see tb-ax1 36707, tb-ax2 36708, and tb-ax3 36709) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊥ → 𝜑)
Theorem	tbwsyl 1723	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 1724	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 1725	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 1726	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 1727	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 1728	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 1729	luk-1 1674 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 1730	luk-2 1675 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 1731	luk-3 1676 derived from the Tarski-Bernays-Wajsberg axioms. This theorem, along with re1luk1 1729 and re1luk2 1730 proves that tbw-ax1 1719, tbw-ax2 1720, tbw-ax3 1721, and tbw-ax4 1722, 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 1732	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 1660 has 21 symbols, sans parentheses. See merco2 1755 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 1733	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (⊥ → 𝜒))
Theorem	retbwax4 1734	tbw-ax4 1722 rederived from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊥ → 𝜑)
Theorem	retbwax2 1735	tbw-ax2 1720 rederived from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	merco1lem2 1736	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (((𝜓 → 𝜏) → (𝜑 → ⊥)) → 𝜒))
Theorem	merco1lem3 1737	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑))
Theorem	merco1lem4 1738	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
Theorem	merco1lem5 1739	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑 → 𝜏))
Theorem	merco1lem6 1740	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜒 → (𝜑 → 𝜓)))
Theorem	merco1lem7 1741	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜓) → 𝜓))
Theorem	retbwax3 1742	tbw-ax3 1721 rederived from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	merco1lem8 1743	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ((𝜓 → (𝜓 → 𝜒)) → (𝜓 → 𝜒)))
Theorem	merco1lem9 1744	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
Theorem	merco1lem10 1745	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (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 1746	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → ⊥) → 𝜓))
Theorem	merco1lem12 1747	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜓))
Theorem	merco1lem13 1748	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → 𝜏) → (𝜑 → 𝜏))
Theorem	merco1lem14 1749	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) → (𝜑 → 𝜒))
Theorem	merco1lem15 1750	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))
Theorem	merco1lem16 1751	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜏) → ((𝜑 → 𝜒) → 𝜏))
Theorem	merco1lem17 1752	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (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 1753	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1732. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜓 → 𝜑) → (𝜓 → 𝜒)))
Theorem	retbwax1 1754	tbw-ax1 1719 rederived from merco1 1732. This theorem, along with retbwax2 1735, retbwax3 1742, and retbwax4 1734, shows that merco1 1732 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 1755	A single axiom for propositional calculus discovered by C. A. Meredith. This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1732. (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 1756	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1755. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))
Theorem	mercolem2 1757	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1755. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))
Theorem	mercolem3 1758	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1755. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))
Theorem	mercolem4 1759	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1755. (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 1760	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1755. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜃 → ((𝜃 → 𝜑) → (𝜏 → (𝜒 → 𝜑))))
Theorem	mercolem6 1761	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1755. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜓 → (𝜑 → 𝜒)))
Theorem	mercolem7 1762	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1755. (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 1763	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1755. (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 1764	tbw-ax1 1719 rederived from merco2 1755. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	re1tbw2 1765	tbw-ax2 1720 rederived from merco2 1755. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	re1tbw3 1766	tbw-ax3 1721 rederived from merco2 1755. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	re1tbw4 1767	tbw-ax4 1722 rederived from merco2 1755. This theorem, along with re1tbw1 1764, re1tbw2 1765, and re1tbw3 1766, shows that merco2 1755, 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 1768	Justification for rb-imdf 1769. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)))
Theorem	rb-imdf 1769	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 1770	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 1771	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 1772	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 1773	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 1774	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 1775	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 1776	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 1777	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 1778	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 1779	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 1780	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 1781	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 1782	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 1783	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 1784	luk-1 1674 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 1785	luk-2 1675 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	re2luk3 1786	luk-3 1676 derived from Russell-Bernays'. This theorem, along with re1axmp 1783, re2luk1 1784, and re2luk2 1785 shows that rb-ax1 1771, rb-ax2 1772, rb-ax3 1773, and rb-ax4 1774, along with anmp 1770, 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 199, modus ponendo tollens I mptnan 1787, modus ponendo tollens II mptxor 1788, and modus tollendo ponens (exclusive-or version) mtpxor 1790. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1790 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 1789. 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 1791 and stoic1b 1792) and thema 3 (stoic3 1795). 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 1793, stoic2b 1794, stoic4a 1796, and stoic4b 1797. 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 317. "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 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 1546. [Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1533 and ancom 464) 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 2688 and related theorems.
Theorem	mptnan 1787	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 1788) 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 1788	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 1789	Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1790, 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 1790	Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1789, 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 1789. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1788, that is, it is exclusive-or df-xor 1531), 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 1788), 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 1791	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 1791 and stoic1b 1792 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 1792	Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1791. (Contributed by David A. Wheeler, 16-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ ¬ 𝜃) → ¬ 𝜑)
Theorem	stoic2a 1793	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, 𝜓⊢ 𝜒; in Metamath we will represent that construct as 𝜑 ∧ 𝜓 → 𝜒. This version a is without the phrase "or both"; see stoic2b 1794 for the version with the phrase "or both". We already have this rule as syldan 600, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	stoic2b 1794	Stoic logic Thema 2 version b. See stoic2a 1793. Version b is with the phrase "or both". We already have this rule as mpd3an3 1482, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	stoic3 1795	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.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
Theorem	stoic4a 1796	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 1797 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
Theorem	stoic4b 1797	Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1796 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ (((𝜒 ∧ 𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 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 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 1828) 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 1799) 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 1799. 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 2217) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as Theorem spw 2053 below). Theorem spw 2053 can be used to prove any instance of sp 2217 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 2217 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 2217 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 2217 as Theorem axc5 39481 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 1814, ax-4 1828, ax-5 1929, ax-6 1986, ax-7 2027, ax-8 2143, and ax-9 2151. 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 2027, ax-8 2143, and ax-9 2151 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 2217, even though (using spw 2053) 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 2174, ax-11 2190, ax-12 2211, and ax-13 2402 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 1557 for use by df-tru 1562. See the comments in that section. In this section, we continue with the first "real" use of it.
1.4.1.1  Existential quantifier
Syntax	wex 1798	Extend wff definition to include the existential quantifier ("there exists").
wff ∃𝑥𝜑
Definition	df-ex 1799	Define existential quantification. ∃𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true". Dual of alex 1845. See also the dual pair alnex 1800 / exnal 1846. Definition of [Margaris] p. 49. (Contributed by NM, 10-Jan-1993.)
⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
Theorem	alnex 1800	Universal quantification of negation is equivalent to negation of existential quantification. Dual of exnal 1846 (but does not depend on ax-4 1828 contrary to it). See also the dual pair df-ex 1799 / alex 1845. Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)