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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | spim 1701 | Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1701 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 1702 | 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 1703 | Deduction version of spime 1704. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) |
Theorem | spime 1704 | 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 1705 | 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 1706 | 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 1707 | 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 1708 | 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 1709 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | cbv2 1710 | 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 1711 | 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 1712 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Theorem | cbvexh 1713 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.) |
Theorem | cbvex 1714 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | chvar 1715 | Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Theorem | equvini 1716 | 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 1717 | A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1716.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.) |
Theorem | nfald 1718 | 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 1719 | 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 1720 | 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 1721 |
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 1733.
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 1796, sbcom2 1940 and sbid2v 1949). Note that our definition is valid even when and are replaced with the same variable, as sbid 1732 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 1944 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 1947. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1843 and sb6 1842. 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 1722 | Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
Theorem | sbbii 1723 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb1 1724 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb2 1725 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ1 1726 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ2 1727 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | stdpc7 1728 | One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1664.) 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 1729 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ12r 1730 | An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | sbequ12a 1731 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid 1732 | 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 1733 | 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 1734 | 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 1735 | 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 1736 | Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.) |
Theorem | sb6x 1737 | Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | nfs1f 1738 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbs1f 1739 | 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 1740 | Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.) |
Theorem | sbequ6 1741 | Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.) |
Theorem | sbt 1742 | A substitution into a theorem remains true. (See chvar 1715 and chvarv 1889 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equsb1 1743 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | equsb2 1744 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbiedh 1745 | Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1748). New proofs should use sbied 1746 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
Theorem | sbied 1746 | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1749). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sbiedv 1747* | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1749). (Contributed by NM, 7-Jan-2017.) |
Theorem | sbieh 1748 | Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1749 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.) |
Theorem | sbie 1749 | 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 | equs5a 1750 | A property related to substitution that unlike equs5 1785 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | equs5e 1751 | A property related to substitution that unlike equs5 1785 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.) |
Theorem | ax11e 1752 | Analogue to ax-11 1469 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.) |
Theorem | ax10oe 1753 | Quantifier Substitution for existential quantifiers. Analogue to ax10o 1678 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.) |
Theorem | drex1 1754 | 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 1755 | 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 1756 | 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 1757 | A version of sb4 1788 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | equs45f 1758 | Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.) |
Theorem | sb6f 1759 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.) |
Theorem | sb5f 1760 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.) |
Theorem | sb4e 1761 | One direction of a simplified definition of substitution that unlike sb4 1788 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2a 1762 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2e 1763 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb3 1764 | If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | nfs1 1765 | If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | sbcof2 1766 | Version of sbco 1919 where is not free in . (Contributed by Jim Kingdon, 28-Dec-2017.) |
Theorem | spimv 1767* | A version of spim 1701 with a distinct variable requirement instead of a bound-variable hypothesis. (Contributed by NM, 5-Aug-1993.) |
Theorem | aev 1768* | A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1770. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | ax16 1769* |
Theorem showing that ax-16 1770 is redundant if ax-17 1491 is included in the
axiom system. The important part of the proof is provided by aev 1768.
See ax16ALT 1815 for an alternate proof that does not require ax-10 1468 or ax-12 1474. This theorem should not be referenced in any proof. Instead, use ax-16 1770 below so that theorems needing ax-16 1770 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
Axiom | ax-16 1770* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1491 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 1491; see theorem ax16 1769. This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1769. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | dveeq2 1771* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | dveeq2or 1772* | Quantifier introduction when one pair of variables is distinct. Like dveeq2 1771 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 1773* | Proof of dvelimf 1968 using dveeq2 1771 (shown as the last hypothesis) instead of ax-12 1474. This shows that ax-12 1474 could be replaced by dveeq2 1771 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) |
Theorem | nd5 1774* | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.) |
Theorem | exlimdv 1775* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
Theorem | ax11v2 1776* | Recovery of ax11o 1778 from ax11v 1783 without using ax-11 1469. The hypothesis is even weaker than ax11v 1783, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1778. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11a2 1777* | Derive ax-11o 1779 from a hypothesis in the form of ax-11 1469. The hypothesis is even weaker than ax-11 1469, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1778. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11o 1778 |
Derivation of set.mm's original ax-11o 1779 from the shorter ax-11 1469 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1770 or ax-17 1491. Normally, ax11o 1778 should be used rather than ax-11o 1779, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.) |
Axiom | ax-11o 1779 |
Axiom ax-11o 1779 ("o" for "old") was the
original version of ax-11 1469,
before it was discovered (in Jan. 2007) that the shorter ax-11 1469 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 1778. This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1778. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | albidv 1780* | Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
Theorem | exbidv 1781* | Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11b 1782 | A bidirectional version of ax-11o 1779. (Contributed by NM, 30-Jun-2006.) |
Theorem | ax11v 1783* | This is a version of ax-11o 1779 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 1784* | Analogue to ax11v 1783 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.) |
Theorem | equs5 1785 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) |
Theorem | equs5or 1786 | Lemma used in proofs of substitution properties. Like equs5 1785 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sb3 1787 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4 1788 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4or 1789 | One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1788 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sb4b 1790 | Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.) |
Theorem | sb4bor 1791 | Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.) |
Theorem | hbsb2 1792 | Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsb2or 1793 | Bound-variable hypothesis builder for substitution. Similar to hbsb2 1792 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sbequilem 1794 | Propositional logic lemma used in the sbequi 1795 proof. (Contributed by Jim Kingdon, 1-Feb-2018.) |
Theorem | sbequi 1795 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.) |
Theorem | sbequ 1796 | 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 1797 | 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 1798 | 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 1799 | Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
Theorem | spsbbi 1800 | Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
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