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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ax11v2 1701* | Recovery of ax11o 1703 from ax11v 1708 without using ax-11 1397. The hypothesis is even weaker than ax11v 1708, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1703. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11a2 1702* | Derive ax-11o 1704 from a hypothesis in the form of ax-11 1397. The hypothesis is even weaker than ax-11 1397, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1703. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11o 1703 |
Derivation of set.mm's original ax-11o 1704 from the shorter ax-11 1397 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1695 or ax-17 1419. Normally, ax11o 1703 should be used rather than ax-11o 1704, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.) |
Axiom | ax-11o 1704 |
Axiom ax-11o 1704 ("o" for "old") was the
original version of ax-11 1397,
before it was discovered (in Jan. 2007) that the shorter ax-11 1397 could
replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of
the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of
[Monk2] p. 105, from which it can be proved
by cases. To understand this
theorem more easily, think of " ..." as informally
meaning "if
and are distinct
variables then..." The
antecedent becomes false if the same variable is substituted for and
, ensuring the
theorem is sound whenever this is the case. In some
later theorems, we call an antecedent of the form a
"distinctor."
This axiom is redundant, as shown by theorem ax11o 1703. This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1703. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | albidv 1705* | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | exbidv 1706* | Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11b 1707 | A bidirectional version of ax-11o 1704. (Contributed by NM, 30-Jun-2006.) |
Theorem | ax11v 1708* | This is a version of ax-11o 1704 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.) |
Theorem | ax11ev 1709* | Analogue to ax11v 1708 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.) |
Theorem | equs5 1710 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) |
Theorem | equs5or 1711 | Lemma used in proofs of substitution properties. Like equs5 1710 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sb3 1712 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4 1713 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4or 1714 | One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1713 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sb4b 1715 | Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.) |
Theorem | sb4bor 1716 | Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.) |
Theorem | hbsb2 1717 | Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsb2or 1718 | Bound-variable hypothesis builder for substitution. Similar to hbsb2 1717 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sbequilem 1719 | Propositional logic lemma used in the sbequi 1720 proof. (Contributed by Jim Kingdon, 1-Feb-2018.) |
Theorem | sbequi 1720 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.) |
Theorem | sbequ 1721 | An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | drsb2 1722 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) |
Theorem | spsbe 1723 | A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.) |
Theorem | spsbim 1724 | Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
Theorem | spsbbi 1725 | Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
Theorem | sbbid 1726 | Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ8 1727 | Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.) |
Theorem | sbft 1728 | Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.) |
Theorem | sbid2h 1729 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid2 1730 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbidm 1731 | An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
Theorem | sb5rf 1732 | Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb6rf 1733 | Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb8h 1734 | Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
Theorem | sb8eh 1735 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.) |
Theorem | sb8 1736 | Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
Theorem | sb8e 1737 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
Theorem | ax16i 1738* | Inference with ax-16 1695 as its conclusion, that doesn't require ax-10 1396, ax-11 1397, or ax-12 1402 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.) |
Theorem | ax16ALT 1739* | Version of ax16 1694 that doesn't require ax-10 1396 or ax-12 1402 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | spv 1740* | Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.) |
Theorem | spimev 1741* | Distinct-variable version of spime 1629. (Contributed by NM, 5-Aug-1993.) |
Theorem | speiv 1742* | Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.) |
Theorem | equvin 1743* | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.) |
Theorem | a16g 1744* | A generalization of axiom ax-16 1695. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | a16gb 1745* | A generalization of axiom ax-16 1695. (Contributed by NM, 5-Aug-1993.) |
Theorem | a16nf 1746* | If there is only one element in the universe, then everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.) |
Theorem | 2albidv 1747* | Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.) |
Theorem | 2exbidv 1748* | Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.) |
Theorem | 3exbidv 1749* | Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.) |
Theorem | 4exbidv 1750* | Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.) |
Theorem | 19.9v 1751* | Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.) |
Theorem | exlimdd 1752 | Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | 19.21v 1753* | Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as in 19.21 1475 via the use of distinct variable conditions combined with ax-17 1419. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g. euf 1905 derived from df-eu 1903. The "f" stands for "not free in" which is less restrictive than "does not occur in." (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimiv 1754* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimivv 1755* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |
Theorem | alrimdv 1756* | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) |
Theorem | nfdv 1757* | Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | 2ax17 1758* | Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.) |
Theorem | alimdv 1759* | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.) |
Theorem | eximdv 1760* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
Theorem | 2alimdv 1761* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.) |
Theorem | 2eximdv 1762* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.) |
Theorem | 19.23v 1763* | Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.) |
Theorem | 19.23vv 1764* | Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.) |
Theorem | sb56 1765* | Two equivalent ways of expressing the proper substitution of for in , when and are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1646. (Contributed by NM, 14-Apr-2008.) |
Theorem | sb6 1766* | Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.) |
Theorem | sb5 1767* | Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.) |
Theorem | sbnv 1768* | Version of sbn 1826 where and are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.) |
Theorem | sbanv 1769* | Version of sban 1829 where and are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Theorem | sborv 1770* | Version of sbor 1828 where and are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.) |
Theorem | sbi1v 1771* | Forward direction of sbimv 1773. (Contributed by Jim Kingdon, 25-Dec-2017.) |
Theorem | sbi2v 1772* | Reverse direction of sbimv 1773. (Contributed by Jim Kingdon, 18-Jan-2018.) |
Theorem | sbimv 1773* | Intuitionistic proof of sbim 1827 where and are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.) |
Theorem | sblimv 1774* | Version of sblim 1831 where and are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.) |
Theorem | pm11.53 1775* | Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
Theorem | exlimivv 1776* | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.) |
Theorem | exlimdvv 1777* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |
Theorem | exlimddv 1778* | Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.) |
Theorem | 19.27v 1779* | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.) |
Theorem | 19.28v 1780* | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.) |
Theorem | 19.36aiv 1781* | Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.41v 1782* | Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.41vv 1783* | Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.) |
Theorem | 19.41vvv 1784* | Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.) |
Theorem | 19.41vvvv 1785* | Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.) |
Theorem | 19.42v 1786* | Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | exdistr 1787* | Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) |
Theorem | 19.42vv 1788* | Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.) |
Theorem | 19.42vvv 1789* | Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.) |
Theorem | 19.42vvvv 1790* | Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.) |
Theorem | exdistr2 1791* | Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.) |
Theorem | 3exdistr 1792* | Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 4exdistr 1793* | Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) |
Theorem | cbvalv 1794* | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) |
Theorem | cbvexv 1795* | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) |
Theorem | cbval2 1796* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) |
Theorem | cbvex2 1797* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | cbval2v 1798* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) |
Theorem | cbvex2v 1799* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Theorem | cbvald 1800* | Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1893. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) |
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