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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sb5f 1701 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.) |
Theorem | sb4e 1702 | One direction of a simplified definition of substitution that unlike sb4 1729 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2a 1703 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2e 1704 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb3 1705 | If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | nfs1 1706 | If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | sbcof2 1707 | Version of sbco 1858 where is not free in . (Contributed by Jim Kingdon, 28-Dec-2017.) |
Theorem | spimv 1708* | A version of spim 1642 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.) |
Theorem | aev 1709* | A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1711. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | ax16 1710* |
Theorem showing that ax-16 1711 is redundant if ax-17 1435 is included in the
axiom system. The important part of the proof is provided by aev 1709.
See ax16ALT 1755 for an alternate proof that does not require ax-10 1412 or ax-12 1418. This theorem should not be referenced in any proof. Instead, use ax-16 1711 below so that theorems needing ax-16 1711 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
Axiom | ax-16 1711* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1435 to be a
metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p.
16 of the preprint). It apparently does not otherwise appear in the
literature but is easily proved from textbook predicate calculus by
cases. It is a somewhat bizarre axiom since the antecedent is always
false in set theory, but nonetheless it is technically necessary as you
can see from its uses.
This axiom is redundant if we include ax-17 1435; see theorem ax16 1710. This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1710. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | dveeq2 1712* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | dveeq2or 1713* | Quantifier introduction when one pair of variables is distinct. Like dveeq2 1712 but connecting by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.) |
Theorem | dvelimfALT2 1714* | Proof of dvelimf 1907 using dveeq2 1712 (shown as the last hypothesis) instead of ax-12 1418. This shows that ax-12 1418 could be replaced by dveeq2 1712 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) |
Theorem | nd5 1715* | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.) |
Theorem | exlimdv 1716* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
Theorem | ax11v2 1717* | Recovery of ax11o 1719 from ax11v 1724 without using ax-11 1413. The hypothesis is even weaker than ax11v 1724, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1719. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11a2 1718* | Derive ax-11o 1720 from a hypothesis in the form of ax-11 1413. The hypothesis is even weaker than ax-11 1413, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1719. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11o 1719 |
Derivation of set.mm's original ax-11o 1720 from the shorter ax-11 1413 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1711 or ax-17 1435. Normally, ax11o 1719 should be used rather than ax-11o 1720, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.) |
Axiom | ax-11o 1720 |
Axiom ax-11o 1720 ("o" for "old") was the
original version of ax-11 1413,
before it was discovered (in Jan. 2007) that the shorter ax-11 1413 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 1719. This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1719. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | albidv 1721* | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | exbidv 1722* | Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11b 1723 | A bidirectional version of ax-11o 1720. (Contributed by NM, 30-Jun-2006.) |
Theorem | ax11v 1724* | This is a version of ax-11o 1720 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 1725* | Analogue to ax11v 1724 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.) |
Theorem | equs5 1726 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) |
Theorem | equs5or 1727 | Lemma used in proofs of substitution properties. Like equs5 1726 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sb3 1728 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4 1729 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4or 1730 | One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1729 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sb4b 1731 | Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.) |
Theorem | sb4bor 1732 | Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.) |
Theorem | hbsb2 1733 | Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsb2or 1734 | Bound-variable hypothesis builder for substitution. Similar to hbsb2 1733 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sbequilem 1735 | Propositional logic lemma used in the sbequi 1736 proof. (Contributed by Jim Kingdon, 1-Feb-2018.) |
Theorem | sbequi 1736 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.) |
Theorem | sbequ 1737 | 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 1738 | 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 1739 | 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 1740 | Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
Theorem | spsbbi 1741 | Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
Theorem | sbbid 1742 | Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ8 1743 | Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.) |
Theorem | sbft 1744 | 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 1745 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid2 1746 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbidm 1747 | An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
Theorem | sb5rf 1748 | Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb6rf 1749 | Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb8h 1750 | 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 1751 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.) |
Theorem | sb8 1752 | 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 1753 | 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 1754* | Inference with ax-16 1711 as its conclusion, that doesn't require ax-10 1412, ax-11 1413, or ax-12 1418 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 1755* | Version of ax16 1710 that doesn't require ax-10 1412 or ax-12 1418 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | spv 1756* | Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.) |
Theorem | spimev 1757* | Distinct-variable version of spime 1645. (Contributed by NM, 5-Aug-1993.) |
Theorem | speiv 1758* | Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.) |
Theorem | equvin 1759* | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.) |
Theorem | a16g 1760* | A generalization of axiom ax-16 1711. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | a16gb 1761* | A generalization of axiom ax-16 1711. (Contributed by NM, 5-Aug-1993.) |
Theorem | a16nf 1762* | If there is only one element in the universe, then everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.) |
Theorem | 2albidv 1763* | Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.) |
Theorem | 2exbidv 1764* | Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.) |
Theorem | 3exbidv 1765* | Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.) |
Theorem | 4exbidv 1766* | Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.) |
Theorem | 19.9v 1767* | Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.) |
Theorem | exlimdd 1768 | Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | 19.21v 1769* | 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 1491 via the use of distinct variable conditions combined with ax-17 1435. 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 1921 derived from df-eu 1919. The "f" stands for "not free in" which is less restrictive than "does not occur in." (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimiv 1770* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimivv 1771* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |
Theorem | alrimdv 1772* | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) |
Theorem | nfdv 1773* | Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | 2ax17 1774* | Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.) |
Theorem | alimdv 1775* | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.) |
Theorem | eximdv 1776* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
Theorem | 2alimdv 1777* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.) |
Theorem | 2eximdv 1778* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.) |
Theorem | 19.23v 1779* | Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.) |
Theorem | 19.23vv 1780* | Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.) |
Theorem | sb56 1781* | 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 1662. (Contributed by NM, 14-Apr-2008.) |
Theorem | sb6 1782* | 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 1783* | 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 1784* | Version of sbn 1842 where and are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.) |
Theorem | sbanv 1785* | Version of sban 1845 where and are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Theorem | sborv 1786* | Version of sbor 1844 where and are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.) |
Theorem | sbi1v 1787* | Forward direction of sbimv 1789. (Contributed by Jim Kingdon, 25-Dec-2017.) |
Theorem | sbi2v 1788* | Reverse direction of sbimv 1789. (Contributed by Jim Kingdon, 18-Jan-2018.) |
Theorem | sbimv 1789* | Intuitionistic proof of sbim 1843 where and are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.) |
Theorem | sblimv 1790* | Version of sblim 1847 where and are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.) |
Theorem | pm11.53 1791* | Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
Theorem | exlimivv 1792* | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.) |
Theorem | exlimdvv 1793* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |
Theorem | exlimddv 1794* | Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.) |
Theorem | 19.27v 1795* | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.) |
Theorem | 19.28v 1796* | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.) |
Theorem | 19.36aiv 1797* | Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.41v 1798* | Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.41vv 1799* | Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.) |
Theorem | 19.41vvv 1800* | Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.) |
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