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	renicax 1701	A rederivation of nic-ax 1677 from lukshef-ax1 1698, proving that lukshef-ax1 1698 with nic-mp 1675 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 1702	Justification for tbw-negdf 1703. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ (((𝜑 → 𝜓) → ((𝜓 → 𝜑) → ⊥)) → ⊥))
Theorem	tbw-negdf 1703	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 1704	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 1705	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 1706	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 1707	The fourth of four axioms in the Tarski-Bernays-Wajsberg system. This axiom was added to the Tarski-Bernays axiom system (see tb-ax1 34499, tb-ax2 34500, and tb-ax3 34501) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊥ → 𝜑)
Theorem	tbwsyl 1708	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 1709	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 1710	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 1711	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 1712	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 1713	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 1714	luk-1 1659 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 1715	luk-2 1660 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 1716	luk-3 1661 derived from the Tarski-Bernays-Wajsberg axioms. This theorem, along with re1luk1 1714 and re1luk2 1715 proves that tbw-ax1 1704, tbw-ax2 1705, tbw-ax3 1706, and tbw-ax4 1707, 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 1717	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 1645 has 21 symbols, sans parentheses. See merco2 1740 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 1718	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (⊥ → 𝜒))
Theorem	retbwax4 1719	tbw-ax4 1707 rederived from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊥ → 𝜑)
Theorem	retbwax2 1720	tbw-ax2 1705 rederived from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	merco1lem2 1721	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (((𝜓 → 𝜏) → (𝜑 → ⊥)) → 𝜒))
Theorem	merco1lem3 1722	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑))
Theorem	merco1lem4 1723	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
Theorem	merco1lem5 1724	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑 → 𝜏))
Theorem	merco1lem6 1725	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜒 → (𝜑 → 𝜓)))
Theorem	merco1lem7 1726	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜓) → 𝜓))
Theorem	retbwax3 1727	tbw-ax3 1706 rederived from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	merco1lem8 1728	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ((𝜓 → (𝜓 → 𝜒)) → (𝜓 → 𝜒)))
Theorem	merco1lem9 1729	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
Theorem	merco1lem10 1730	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (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 1731	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → ⊥) → 𝜓))
Theorem	merco1lem12 1732	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜓))
Theorem	merco1lem13 1733	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → 𝜏) → (𝜑 → 𝜏))
Theorem	merco1lem14 1734	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) → (𝜑 → 𝜒))
Theorem	merco1lem15 1735	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))
Theorem	merco1lem16 1736	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜏) → ((𝜑 → 𝜒) → 𝜏))
Theorem	merco1lem17 1737	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (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 1738	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜓 → 𝜑) → (𝜓 → 𝜒)))
Theorem	retbwax1 1739	tbw-ax1 1704 rederived from merco1 1717. This theorem, along with retbwax2 1720, retbwax3 1727, and retbwax4 1719, shows that merco1 1717 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 1740	A single axiom for propositional calculus discovered by C. A. Meredith. This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1717. (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 1741	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1740. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))
Theorem	mercolem2 1742	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1740. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))
Theorem	mercolem3 1743	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1740. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))
Theorem	mercolem4 1744	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1740. (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 1745	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1740. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜃 → ((𝜃 → 𝜑) → (𝜏 → (𝜒 → 𝜑))))
Theorem	mercolem6 1746	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1740. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜓 → (𝜑 → 𝜒)))
Theorem	mercolem7 1747	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1740. (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 1748	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1740. (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 1749	tbw-ax1 1704 rederived from merco2 1740. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	re1tbw2 1750	tbw-ax2 1705 rederived from merco2 1740. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	re1tbw3 1751	tbw-ax3 1706 rederived from merco2 1740. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	re1tbw4 1752	tbw-ax4 1707 rederived from merco2 1740. This theorem, along with re1tbw1 1749, re1tbw2 1750, and re1tbw3 1751, shows that merco2 1740, 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 1753	Justification for rb-imdf 1754. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)))
Theorem	rb-imdf 1754	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 1755	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 1756	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 1757	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 1758	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 1759	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 1760	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 1761	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 1762	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 1763	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 1764	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 1765	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 1766	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 1767	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 1768	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 1769	luk-1 1659 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 1770	luk-2 1660 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	re2luk3 1771	luk-3 1661 derived from Russell-Bernays'. This theorem, along with re1axmp 1768, re2luk1 1769, and re2luk2 1770 shows that rb-ax1 1756, rb-ax2 1757, rb-ax3 1758, and rb-ax4 1759, along with anmp 1755, 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 196, modus ponendo tollens I mptnan 1772, modus ponendo tollens II mptxor 1773, and modus tollendo ponens (exclusive-or version) mtpxor 1775. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1775 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 1774. 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 1776 and stoic1b 1777) and thema 3 (stoic3 1780). 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 1778, stoic2b 1779, stoic4a 1781, and stoic4b 1782. 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 314. "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 1521. [Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1506 and ancom 460) 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 2664 and related theorems.
Theorem	mptnan 1772	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 1773) 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 1773	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 1774	Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1775, 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 1775	Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1774, 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 1774. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1773, that is, it is exclusive-or df-xor 1504), 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 1773), 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 1776	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 1776 and stoic1b 1777 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 1777	Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1776. (Contributed by David A. Wheeler, 16-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ ¬ 𝜃) → ¬ 𝜑)
Theorem	stoic2a 1778	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 1779 for the version with the phrase "or both". We already have this rule as syldan 590, 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 1779	Stoic logic Thema 2 version b. See stoic2a 1778. Version b is with the phrase "or both". We already have this rule as mpd3an3 1460, 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 1780	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 1781	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 1782 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 1782	Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1781 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 us to talk about individual objects in a domain of discourse (which for us will be the universe of all sets, so we call them "setvar variables") and make true/false statements about predicates, which are relationships between objects, such as whether or not two objects are equal. In addition, we introduce universal quantification ("for all", e.g., ax-4 1813) 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 1784) 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 1784. 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 2178) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as Theorem spw 2038 below). Theorem spw 2038 can be used to prove any instance of sp 2178 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 2178 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 2178 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 2178 as Theorem axc5 36834 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 1799, ax-4 1813, ax-5 1914, ax-6 1972, ax-7 2012, ax-8 2110, and ax-9 2118. 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 2012, ax-8 2110, and ax-9 2118 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 2178, even though (using spw 2038) 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 2139, ax-11 2156, ax-12 2173, and ax-13 2372 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 1537 for use by df-tru 1542. 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 1783	Extend wff definition to include the existential quantifier ("there exists").
wff ∃𝑥𝜑
Definition	df-ex 1784	Define existential quantification. ∃𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true". Dual of alex 1829. See also the dual pair alnex 1785 / exnal 1830. Definition of [Margaris] p. 49. (Contributed by NM, 10-Jan-1993.)
⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
Theorem	alnex 1785	Universal quantification of negation is equivalent to negation of existential quantification. Dual of exnal 1830 (but does not depend on ax-4 1813 contrary to it). See also the dual pair df-ex 1784 / alex 1829. Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
Theorem	eximal 1786	An equivalence between an implication with an existentially quantified antecedent and an implication with a universally quantified consequent. An interesting case is when the same formula is substituted for both 𝜑 and 𝜓, since then both implications express a type of nonfreeness. See also alimex 1834. (Contributed by BJ, 12-May-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
1.4.1.2  Nonfreeness predicate
Syntax	wnf 1787	Extend wff definition to include the not-free predicate.
wff Ⅎ𝑥𝜑
Definition	df-nf 1788	Define the not-free predicate for wffs. This is read "𝑥 is not free in 𝜑". Not-free means that the value of 𝑥 cannot affect the value of 𝜑, e.g., any occurrence of 𝑥 in 𝜑 is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 2266). An example of where this is used is stdpc5 2204. See nf5 2282 for an alternate definition which involves nested quantifiers on the same variable. Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition. To be precise, our definition really means "effectively not free", because it is slightly less restrictive than the usual textbook definition for "not free" (which considers syntactic freedom). For example, 𝑥 is effectively not free in the formula 𝑥 = 𝑥 (even though 𝑥 is syntactically free in it, so would be considered free in the usual textbook definition) because the value of 𝑥 in the formula 𝑥 = 𝑥 does not affect the truth of that formula (and thus substitutions will not change the result), see nfequid 2017. This definition of "not free" tightly ties to the quantifier ∀𝑥. At this state (no axioms restricting quantifiers yet) "nonfree" appears quite arbitrary. Its intended semantics expresses single-valuedness (constness) across a parameter, but is only evolved as much as later axioms assign properties to quantifiers. It seems the definition here is best suited in situations, where axioms are only partially in effect. In particular, this definition more easily carries over to other logic models with weaker axiomization. The reverse implication of the definiens (the right hand side of the biconditional) always holds, see 19.2 1981. This predicate only applies to wffs. See df-nfc 2888 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 24-Sep-2016.) Convert to definition. (Revised by BJ, 6-May-2019.)
⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	nf2 1789	Alternate definition of nonfreeness. (Contributed by BJ, 16-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
Theorem	nf3 1790	Alternate definition of nonfreeness. (Contributed by BJ, 16-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
Theorem	nf4 1791	Alternate definition of nonfreeness. This definition uses only primitive symbols (→ , ¬ , ∀). (Contributed by BJ, 16-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
Theorem	nfi 1792	Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Wolf Lammen, 15-Sep-2021.)
⊢ (∃𝑥𝜑 → ∀𝑥𝜑)    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfri 1793	Consequence of the definition of not-free. (Contributed by Wolf Lammen, 16-Sep-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
Theorem	nfd 1794	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)
⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nfrd 1795	Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
Theorem	nftht 1796	Closed form of nfth 1805. (Contributed by Wolf Lammen, 19-Aug-2018.) (Proof shortened by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 3-Sep-2022.)
⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	nfntht 1797	Closed form of nfnth 1806. (Contributed by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 4-Sep-2022.)
⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	nfntht2 1798	Closed form of nfnth 1806. (Contributed by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 4-Sep-2022.)
⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)
1.4.2  Rule scheme ax-gen (Generalization)
Axiom	ax-gen 1799	Rule of (universal) generalization. In our axiomatization, this is the only postulated (that is, axiomatic) rule of inference of predicate calculus (together with the rule of modus ponens ax-mp 5 of propositional calculus). See, e.g., Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved 𝑥 = 𝑥, then we can conclude ∀𝑥𝑥 = 𝑥 or even ∀𝑦𝑥 = 𝑥. Theorem altru 1811 shows the special case ∀𝑥⊤. The converse rule of inference spi 2179 (universal instantiation, or universal specialization) shows that we can also go the other way: in other words, we can add or remove universal quantifiers from the beginning of any theorem as required. Note that the closed form (𝜑 → ∀𝑥𝜑) need not hold (but may hold in special cases, see ax-5 1914). (Contributed by NM, 3-Jan-1993.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥𝜑
Theorem	gen2 1800	Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥∀𝑦𝜑