Home | Intuitionistic Logic Explorer Theorem List (p. 18 of 129) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nfald 1701 | 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 1702 | 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 1703 | 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 1704 |
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 1716.
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 1779, sbcom2 1923 and sbid2v 1932). Note that our definition is valid even when and are replaced with the same variable, as sbid 1715 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 1927 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 1930. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1826 and sb6 1825. 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 1705 | Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
Theorem | sbbii 1706 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb1 1707 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb2 1708 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ1 1709 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ2 1710 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | stdpc7 1711 | One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1647.) 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 1712 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ12r 1713 | An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | sbequ12a 1714 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid 1715 | 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 1716 | 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 1717 | 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 1718 | 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 1719 | Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.) |
Theorem | sb6x 1720 | Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | nfs1f 1721 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbs1f 1722 | 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 1723 | Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.) |
Theorem | sbequ6 1724 | Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.) |
Theorem | sbt 1725 | A substitution into a theorem remains true. (See chvar 1698 and chvarv 1872 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equsb1 1726 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | equsb2 1727 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbiedh 1728 | Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1731). New proofs should use sbied 1729 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
Theorem | sbied 1729 | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1732). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sbiedv 1730* | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1732). (Contributed by NM, 7-Jan-2017.) |
Theorem | sbieh 1731 | Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1732 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.) |
Theorem | sbie 1732 | 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 1733 | A property related to substitution that unlike equs5 1768 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | equs5e 1734 | A property related to substitution that unlike equs5 1768 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.) |
Theorem | ax11e 1735 | Analogue to ax-11 1452 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.) |
Theorem | ax10oe 1736 | Quantifier Substitution for existential quantifiers. Analogue to ax10o 1661 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.) |
Theorem | drex1 1737 | 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 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, 5-Aug-1993.) |
Theorem | exdistrfor 1739 | 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 1740 | A version of sb4 1771 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | equs45f 1741 | Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.) |
Theorem | sb6f 1742 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.) |
Theorem | sb5f 1743 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.) |
Theorem | sb4e 1744 | One direction of a simplified definition of substitution that unlike sb4 1771 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2a 1745 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2e 1746 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb3 1747 | If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | nfs1 1748 | If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | sbcof2 1749 | Version of sbco 1902 where is not free in . (Contributed by Jim Kingdon, 28-Dec-2017.) |
Theorem | spimv 1750* | A version of spim 1684 with a distinct variable requirement instead of a bound-variable hypothesis. (Contributed by NM, 5-Aug-1993.) |
Theorem | aev 1751* | A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1753. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | ax16 1752* |
Theorem showing that ax-16 1753 is redundant if ax-17 1474 is included in the
axiom system. The important part of the proof is provided by aev 1751.
See ax16ALT 1798 for an alternate proof that does not require ax-10 1451 or ax-12 1457. This theorem should not be referenced in any proof. Instead, use ax-16 1753 below so that theorems needing ax-16 1753 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
Axiom | ax-16 1753* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1474 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 1474; see theorem ax16 1752. This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1752. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | dveeq2 1754* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | dveeq2or 1755* | Quantifier introduction when one pair of variables is distinct. Like dveeq2 1754 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 1756* | Proof of dvelimf 1951 using dveeq2 1754 (shown as the last hypothesis) instead of ax-12 1457. This shows that ax-12 1457 could be replaced by dveeq2 1754 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) |
Theorem | nd5 1757* | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.) |
Theorem | exlimdv 1758* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
Theorem | ax11v2 1759* | Recovery of ax11o 1761 from ax11v 1766 without using ax-11 1452. The hypothesis is even weaker than ax11v 1766, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1761. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11a2 1760* | Derive ax-11o 1762 from a hypothesis in the form of ax-11 1452. The hypothesis is even weaker than ax-11 1452, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1761. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11o 1761 |
Derivation of set.mm's original ax-11o 1762 from the shorter ax-11 1452 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1753 or ax-17 1474. Normally, ax11o 1761 should be used rather than ax-11o 1762, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.) |
Axiom | ax-11o 1762 |
Axiom ax-11o 1762 ("o" for "old") was the
original version of ax-11 1452,
before it was discovered (in Jan. 2007) that the shorter ax-11 1452 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 1761. This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1761. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | albidv 1763* | Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
Theorem | exbidv 1764* | Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11b 1765 | A bidirectional version of ax-11o 1762. (Contributed by NM, 30-Jun-2006.) |
Theorem | ax11v 1766* | This is a version of ax-11o 1762 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 1767* | Analogue to ax11v 1766 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.) |
Theorem | equs5 1768 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) |
Theorem | equs5or 1769 | Lemma used in proofs of substitution properties. Like equs5 1768 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sb3 1770 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4 1771 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4or 1772 | One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1771 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sb4b 1773 | Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.) |
Theorem | sb4bor 1774 | Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.) |
Theorem | hbsb2 1775 | Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsb2or 1776 | Bound-variable hypothesis builder for substitution. Similar to hbsb2 1775 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Theorem | sbequilem 1777 | Propositional logic lemma used in the sbequi 1778 proof. (Contributed by Jim Kingdon, 1-Feb-2018.) |
Theorem | sbequi 1778 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.) |
Theorem | sbequ 1779 | 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 1780 | 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 1781 | 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 1782 | Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
Theorem | spsbbi 1783 | Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
Theorem | sbbidh 1784 | Deduction substituting both sides of a biconditional. New proofs should use sbbid 1785 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | sbbid 1785 | Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) |
Theorem | sbequ8 1786 | Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.) |
Theorem | sbft 1787 | 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 1788 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid2 1789 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbidm 1790 | An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
Theorem | sb5rf 1791 | Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb6rf 1792 | Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb8h 1793 | 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 1794 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.) |
Theorem | sb8 1795 | 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 1796 | 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 1797* | Inference with ax-16 1753 as its conclusion, that doesn't require ax-10 1451, ax-11 1452, or ax-12 1457 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 1798* | Version of ax16 1752 that doesn't require ax-10 1451 or ax-12 1457 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | spv 1799* | Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.) |
Theorem | spimev 1800* | Distinct-variable version of spime 1687. (Contributed by NM, 5-Aug-1993.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |