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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nfnae 1701 | All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbnaes 1702 | Rule that applies hbnae 1700 to antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | naecoms 1703 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) |
Theorem | equs4 1704 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) |
Theorem | equsalh 1705 | A useful equivalence related to substitution. New proofs should use equsal 1706 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Theorem | equsal 1706 | A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) |
Theorem | equsex 1707 | A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | equsexd 1708 | Deduction form of equsex 1707. (Contributed by Jim Kingdon, 29-Dec-2017.) |
Theorem | dral1 1709 | 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, 24-Nov-1994.) |
Theorem | dral2 1710 | 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 | drex2 1711 | 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 | drnf1 1712 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | drnf2 1713 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | spimth 1714 | Closed theorem form of spim 1717. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.) |
Theorem | spimt 1715 | Closed theorem form of spim 1717. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.) |
Theorem | spimh 1716 | Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1717 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.) |
Theorem | spim 1717 | Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1717 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.) |
Theorem | spimeh 1718 | Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.) |
Theorem | spimed 1719 | Deduction version of spime 1720. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) |
Theorem | spime 1720 | Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) |
Theorem | cbv3 1721 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
Theorem | cbv3h 1722 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
Theorem | cbv1 1723 | Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) |
Theorem | cbv1h 1724 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.) |
Theorem | cbv2h 1725 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | cbv2 1726 | Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) |
Theorem | cbvalh 1727 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | cbval 1728 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Theorem | cbvexh 1729 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.) |
Theorem | cbvex 1730 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | chvar 1731 | Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Theorem | equvini 1732 | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require to be distinct from and (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equveli 1733 | A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1732.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.) |
Theorem | nfald 1734 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) |
Theorem | nfexd 1735 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.) |
Syntax | wsb 1736 | Extend wff definition to include proper substitution (read "the wff that results when is properly substituted for in wff "). (Contributed by NM, 24-Jan-2006.) |
Definition | df-sb 1737 |
Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the
preprint). For our notation, we use to mean "the wff
that results when
is properly substituted for in the wff
." We
can also use in
place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 1749.
Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, " is the wff that results when is properly substituted for in ." For example, if the original is , then is , from which we obtain that is . So what exactly does mean? Curry's notation solves this problem. In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1813, sbcom2 1963 and sbid2v 1972). Note that our definition is valid even when and are replaced with the same variable, as sbid 1748 shows. We achieve this by having free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 1967 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1970. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1860 and sb6 1859. In classical logic, another possible definition is but we do not have an intuitionistic proof that this is equivalent. There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 5-Aug-1993.) |
Theorem | sbimi 1738 | Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
Theorem | sbbii 1739 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb1 1740 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb2 1741 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ1 1742 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ2 1743 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | stdpc7 1744 | One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1680.) Translated to traditional notation, it can be read: " , , , provided that is free for in , ." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.) |
Theorem | sbequ12 1745 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ12r 1746 | An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | sbequ12a 1747 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid 1748 | An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | stdpc4 1749 | The specialization axiom of standard predicate calculus. It states that if a statement holds for all , then it also holds for the specific case of (properly) substituted for . Translated to traditional notation, it can be read: " , provided that is free for in ." Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbh 1750 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.) |
Theorem | sbf 1751 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sbf2 1752 | Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.) |
Theorem | sb6x 1753 | Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | nfs1f 1754 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbs1f 1755 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sbequ5 1756 | Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.) |
Theorem | sbequ6 1757 | Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.) |
Theorem | sbt 1758 | A substitution into a theorem remains true. (See chvar 1731 and chvarv 1910 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equsb1 1759 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | equsb2 1760 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbiedh 1761 | Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1764). New proofs should use sbied 1762 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
Theorem | sbied 1762 | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1765). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sbiedv 1763* | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1765). (Contributed by NM, 7-Jan-2017.) |
Theorem | sbieh 1764 | Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1765 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.) |
Theorem | sbie 1765 | Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 30-Apr-2018.) |
Theorem | sbiev 1766* | Conversion of implicit substitution to explicit substitution. Version of sbie 1765 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.) |
Theorem | equs5a 1767 | A property related to substitution that unlike equs5 1802 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | equs5e 1768 | A property related to substitution that unlike equs5 1802 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.) |
Theorem | ax11e 1769 | Analogue to ax-11 1485 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.) |
Theorem | ax10oe 1770 | Quantifier Substitution for existential quantifiers. Analogue to ax10o 1694 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.) |
Theorem | drex1 1771 | 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.) (Revised by NM, 3-Feb-2015.) |
Theorem | drsb1 1772 | 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, 5-Aug-1993.) |
Theorem | exdistrfor 1773 | Distribution of existential quantifiers, with a bound-variable hypothesis saying that is not free in , but can be free in (and there is no distinct variable condition on and ). (Contributed by Jim Kingdon, 25-Feb-2018.) |
Theorem | sb4a 1774 | A version of sb4 1805 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | equs45f 1775 | Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.) |
Theorem | sb6f 1776 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.) |
Theorem | sb5f 1777 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.) |
Theorem | sb4e 1778 | One direction of a simplified definition of substitution that unlike sb4 1805 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2a 1779 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2e 1780 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb3 1781 | If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | nfs1 1782 | If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | sbcof2 1783 | Version of sbco 1942 where is not free in . (Contributed by Jim Kingdon, 28-Dec-2017.) |
Theorem | spimv 1784* | A version of spim 1717 with a distinct variable requirement instead of a bound-variable hypothesis. (Contributed by NM, 5-Aug-1993.) |
Theorem | aev 1785* | A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1787. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | ax16 1786* |
Theorem showing that ax-16 1787 is redundant if ax-17 1507 is included in the
axiom system. The important part of the proof is provided by aev 1785.
See ax16ALT 1832 for an alternate proof that does not require ax-10 1484 or ax-12 1490. This theorem should not be referenced in any proof. Instead, use ax-16 1787 below so that theorems needing ax-16 1787 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
Axiom | ax-16 1787* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1507 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 1507; see theorem ax16 1786. This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1786. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | dveeq2 1788* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | dveeq2or 1789* | Quantifier introduction when one pair of variables is distinct. Like dveeq2 1788 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 1790* | Proof of dvelimf 1991 using dveeq2 1788 (shown as the last hypothesis) instead of ax-12 1490. This shows that ax-12 1490 could be replaced by dveeq2 1788 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) |
Theorem | nd5 1791* | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.) |
Theorem | exlimdv 1792* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
Theorem | ax11v2 1793* | Recovery of ax11o 1795 from ax11v 1800 without using ax-11 1485. The hypothesis is even weaker than ax11v 1800, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1795. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11a2 1794* | Derive ax-11o 1796 from a hypothesis in the form of ax-11 1485. The hypothesis is even weaker than ax-11 1485, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1795. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11o 1795 |
Derivation of set.mm's original ax-11o 1796 from the shorter ax-11 1485 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1787 or ax-17 1507. Normally, ax11o 1795 should be used rather than ax-11o 1796, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.) |
Axiom | ax-11o 1796 |
Axiom ax-11o 1796 ("o" for "old") was the
original version of ax-11 1485,
before it was discovered (in Jan. 2007) that the shorter ax-11 1485 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 1795. This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1795. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | albidv 1797* | Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
Theorem | exbidv 1798* | Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11b 1799 | A bidirectional version of ax-11o 1796. (Contributed by NM, 30-Jun-2006.) |
Theorem | ax11v 1800* | This is a version of ax-11o 1796 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.) |
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