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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lukshef-ax1 1701 |
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 1678 for its constructions. Here, the axiom is proved as a substitution instance of nic-ax 1680. (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 1702 | Lemma for renicax 1704. (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 1703 | Lemma for renicax 1704. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜏 ⊼ 𝜃) ⊼ ((𝜃 ⊼ 𝜏) ⊼ (𝜃 ⊼ 𝜏))) | ||
Theorem | renicax 1704 | A rederivation of nic-ax 1680 from lukshef-ax1 1701, proving that lukshef-ax1 1701 with nic-mp 1678 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))))) | ||
Theorem | tbw-bijust 1705 | Justification for tbw-negdf 1706. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) ↔ (((𝜑 → 𝜓) → ((𝜓 → 𝜑) → ⊥)) → ⊥)) | ||
Theorem | tbw-negdf 1706 | 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 1707 | 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 1708 | 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 1709 | 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 1710 |
The fourth of four axioms in the Tarski-Bernays-Wajsberg system.
This axiom was added to the Tarski-Bernays axiom system (see tb-ax1 34210, tb-ax2 34211, and tb-ax3 34212) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊥ → 𝜑) | ||
Theorem | tbwsyl 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 → 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | tbwlem1 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 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | tbwlem2 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 → 𝜒) → 𝜃) → (𝜓 → 𝜃))) | ||
Theorem | tbwlem3 1714 | 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 1715 | 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 1716 | 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 1717 | luk-1 1662 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 1718 | luk-2 1663 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 1719 |
luk-3 1664 derived from the Tarski-Bernays-Wajsberg
axioms.
This theorem, along with re1luk1 1717 and re1luk2 1718 proves that tbw-ax1 1707, tbw-ax2 1708, tbw-ax3 1709, and tbw-ax4 1710, 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)) | ||
Theorem | merco1 1720 |
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 1648 has 21 symbols, sans parentheses. See merco2 1743 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 1721 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (⊥ → 𝜒)) | ||
Theorem | retbwax4 1722 | tbw-ax4 1710 rederived from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊥ → 𝜑) | ||
Theorem | retbwax2 1723 | tbw-ax2 1708 rederived from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Theorem | merco1lem2 1724 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → (((𝜓 → 𝜏) → (𝜑 → ⊥)) → 𝜒)) | ||
Theorem | merco1lem3 1725 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑)) | ||
Theorem | merco1lem4 1726 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)) | ||
Theorem | merco1lem5 1727 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑 → 𝜏)) | ||
Theorem | merco1lem6 1728 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜒 → (𝜑 → 𝜓))) | ||
Theorem | merco1lem7 1729 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜓) → 𝜓)) | ||
Theorem | retbwax3 1730 | tbw-ax3 1709 rederived from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) | ||
Theorem | merco1lem8 1731 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → ((𝜓 → (𝜓 → 𝜒)) → (𝜓 → 𝜒))) | ||
Theorem | merco1lem9 1732 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)) | ||
Theorem | merco1lem10 1733 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (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 1734 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → ⊥) → 𝜓)) | ||
Theorem | merco1lem12 1735 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜓)) | ||
Theorem | merco1lem13 1736 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → 𝜏) → (𝜑 → 𝜏)) | ||
Theorem | merco1lem14 1737 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) → (𝜑 → 𝜒)) | ||
Theorem | merco1lem15 1738 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))) | ||
Theorem | merco1lem16 1739 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜏) → ((𝜑 → 𝜒) → 𝜏)) | ||
Theorem | merco1lem17 1740 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (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 1741 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1720. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜓 → 𝜑) → (𝜓 → 𝜒))) | ||
Theorem | retbwax1 1742 |
tbw-ax1 1707 rederived from merco1 1720.
This theorem, along with retbwax2 1723, retbwax3 1730, and retbwax4 1722, shows that merco1 1720 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))) | ||
Theorem | merco2 1743 |
A single axiom for propositional calculus discovered by C. A. Meredith.
This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1720. (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 1744 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1743. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒))) | ||
Theorem | mercolem2 1745 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1743. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑))) | ||
Theorem | mercolem3 1746 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1743. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | mercolem4 1747 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1743. (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 1748 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1743. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 → ((𝜃 → 𝜑) → (𝜏 → (𝜒 → 𝜑)))) | ||
Theorem | mercolem6 1749 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1743. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | mercolem7 1750 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1743. (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 1751 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1743. (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 1752 | tbw-ax1 1707 rederived from merco2 1743. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||
Theorem | re1tbw2 1753 | tbw-ax2 1708 rederived from merco2 1743. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Theorem | re1tbw3 1754 | tbw-ax3 1709 rederived from merco2 1743. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) | ||
Theorem | re1tbw4 1755 |
tbw-ax4 1710 rederived from merco2 1743.
This theorem, along with re1tbw1 1752, re1tbw2 1753, and re1tbw3 1754, shows that merco2 1743, 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) | ||
Theorem | rb-bijust 1756 | Justification for rb-imdf 1757. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑))) | ||
Theorem | rb-imdf 1757 | 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 1758 | 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 1759 | 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 1760 | 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 1761 | 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 1762 | 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 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.) |
⊢ (¬ 𝜓 ∨ 𝜒) & ⊢ (𝜑 ∨ 𝜓) ⇒ ⊢ (𝜑 ∨ 𝜒) | ||
Theorem | rblem1 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 ∨ 𝜓) & ⊢ (¬ 𝜒 ∨ 𝜃) ⇒ ⊢ (¬ (𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)) | ||
Theorem | rblem2 1765 | 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 1766 | 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 1767 | 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 1768 | 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 1769 | 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 1770 | 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 1771 | 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 1772 | luk-1 1662 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 1773 | luk-2 1663 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((¬ 𝜑 → 𝜑) → 𝜑) | ||
Theorem | re2luk3 1774 |
luk-3 1664 derived from Russell-Bernays'.
This theorem, along with re1axmp 1771, re2luk1 1772, and re2luk2 1773 shows that rb-ax1 1759, rb-ax2 1760, rb-ax3 1761, and rb-ax4 1762, along with anmp 1758, 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)) | ||
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 1775, modus ponendo tollens II mptxor 1776, and modus tollendo ponens (exclusive-or version) mtpxor 1778. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1778 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 1777. 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 1779 and stoic1b 1780) and thema 3 (stoic3 1783). 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 1781, stoic2b 1782, stoic4a 1784, and stoic4b 1785. 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 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 1524. [Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1509 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 2665 and related theorems. | ||
Theorem | mptnan 1775 | 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 1776) 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 1776 | 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 1777 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1778, 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 1778 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1777, 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 1777. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1776, that is, it is exclusive-or df-xor 1507), 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 1776), 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 1779 |
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 1779 and stoic1b 1780 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 1780 | Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1779. (Contributed by David A. Wheeler, 16-Feb-2019.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜃) ⇒ ⊢ ((𝜓 ∧ ¬ 𝜃) → ¬ 𝜑) | ||
Theorem | stoic2a 1781 | 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 1782 for the version with the phrase "or both". We already have this rule as syldan 594, 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 1782 | Stoic logic Thema 2 version b. See stoic2a 1781. Version b is with the phrase "or both". We already have this rule as mpd3an3 1463, 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 1783 | 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 1784 |
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 1785 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 1785 | Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1784 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ (((𝜒 ∧ 𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | ||
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 1816) 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 1787) 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 1787. 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 2184) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as Theorem spw 2046 below). Theorem spw 2046 can be used to prove any instance of sp 2184 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 2184 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 2184 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 2184 as Theorem axc5 36530 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 1802, ax-4 1816, ax-5 1917, ax-6 1975, ax-7 2020, ax-8 2116, and ax-9 2124. 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 2020, ax-8 2116, and ax-9 2124 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 2184, even though (using spw 2046) 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 2145, ax-11 2162, ax-12 2179, 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. | ||
The universal quantifier was introduced above in wal 1540 for use by df-tru 1545. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Syntax | wex 1786 | Extend wff definition to include the existential quantifier ("there exists"). |
wff ∃𝑥𝜑 | ||
Definition | df-ex 1787 | Define existential quantification. ∃𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true". Dual of alex 1832. See also the dual pair alnex 1788 / exnal 1833. Definition of [Margaris] p. 49. (Contributed by NM, 10-Jan-1993.) |
⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | ||
Theorem | alnex 1788 | Universal quantification of negation is equivalent to negation of existential quantification. Dual of exnal 1833 (but does not depend on ax-4 1816 contrary to it). See also the dual pair df-ex 1787 / alex 1832. Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.) |
⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | ||
Theorem | eximal 1789 | 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 1837. (Contributed by BJ, 12-May-2019.) |
⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)) | ||
Syntax | wnf 1790 | Extend wff definition to include the not-free predicate. |
wff Ⅎ𝑥𝜑 | ||
Definition | df-nf 1791 |
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 2271). An example of where this is used is
stdpc5 2210. See nf5 2286 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 2025. 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 1986. This predicate only applies to wffs. See df-nfc 2881 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 1792 | Alternate definition of nonfreeness. (Contributed by BJ, 16-Sep-2021.) |
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) | ||
Theorem | nf3 1793 | Alternate definition of nonfreeness. (Contributed by BJ, 16-Sep-2021.) |
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) | ||
Theorem | nf4 1794 | Alternate definition of nonfreeness. This definition uses only primitive symbols (→ , ¬ , ∀). (Contributed by BJ, 16-Sep-2021.) |
⊢ (Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑)) | ||
Theorem | nfi 1795 | Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Wolf Lammen, 15-Sep-2021.) |
⊢ (∃𝑥𝜑 → ∀𝑥𝜑) ⇒ ⊢ Ⅎ𝑥𝜑 | ||
Theorem | nfri 1796 | Consequence of the definition of not-free. (Contributed by Wolf Lammen, 16-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥𝜑 → ∀𝑥𝜑) | ||
Theorem | nfd 1797 | 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 1798 | Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) | ||
Theorem | nftht 1799 | Closed form of nfth 1808. (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 1800 | Closed form of nfnth 1809. (Contributed by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 4-Sep-2022.) |
⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑) |
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