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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | eeor 1601 | Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.) |
Theorem | a9e 1602 | At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1352 through ax-14 1421 and ax-17 1435, all axioms other than ax-9 1440 are believed to be theorems of free logic, although the system without ax-9 1440 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | a9ev 1603* | At least one individual exists. Weaker version of a9e 1602. (Contributed by NM, 3-Aug-2017.) |
Theorem | ax9o 1604 | An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | equid 1605 |
Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68.
This is often an axiom of equality in textbook systems, but we don't
need it as an axiom since it can be proved from our other axioms.
This proof is similar to Tarski's and makes use of a dummy variable . It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.) |
Theorem | nfequid 1606 | Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.) |
Theorem | stdpc6 1607 | One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1669.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.) |
Theorem | equcomi 1608 | Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.) |
Theorem | equcom 1609 | Commutative law for equality. (Contributed by NM, 20-Aug-1993.) |
Theorem | equcoms 1610 | An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.) |
Theorem | equtr 1611 | A transitive law for equality. (Contributed by NM, 23-Aug-1993.) |
Theorem | equtrr 1612 | A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.) |
Theorem | equtr2 1613 | A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equequ1 1614 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) |
Theorem | equequ2 1615 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) |
Theorem | elequ1 1616 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
Theorem | elequ2 1617 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11i 1618 | Inference that has ax-11 1413 (without ) as its conclusion and 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. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.) |
Theorem | ax10o 1619 |
Show that ax-10o 1620 can be derived from ax-10 1412. An open problem is
whether this theorem can be derived from ax-10 1412 and the others when
ax-11 1413 is replaced with ax-11o 1720. See theorem ax10 1621
for the
rederivation of ax-10 1412 from ax10o 1619.
Normally, ax10o 1619 should be used rather than ax-10o 1620, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) |
Axiom | ax-10o 1620 |
Axiom ax-10o 1620 ("o" for "old") was the
original version of ax-10 1412,
before it was discovered (in May 2008) that the shorter ax-10 1412 could
replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of
the preprint).
This axiom is redundant, as shown by theorem ax10o 1619. Normally, ax10o 1619 should be used rather than ax-10o 1620, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | ax10 1621 |
Rederivation of ax-10 1412 from original version ax-10o 1620. See theorem
ax10o 1619 for the derivation of ax-10o 1620 from ax-10 1412.
This theorem should not be referenced in any proof. Instead, use ax-10 1412 above so that uses of ax-10 1412 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.) |
Theorem | hbae 1622 | All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | nfae 1623 | All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbaes 1624 | Rule that applies hbae 1622 to antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | hbnae 1625 | All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | nfnae 1626 | All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbnaes 1627 | Rule that applies hbnae 1625 to antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | naecoms 1628 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) |
Theorem | equs4 1629 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) |
Theorem | equsalh 1630 | A useful equivalence related to substitution. New proofs should use equsal 1631 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Theorem | equsal 1631 | 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 1632 | A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | equsexd 1633 | Deduction form of equsex 1632. (Contributed by Jim Kingdon, 29-Dec-2017.) |
Theorem | dral1 1634 | 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 1635 | 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 1636 | 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 1637 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | drnf2 1638 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | spimth 1639 | Closed theorem form of spim 1642. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.) |
Theorem | spimt 1640 | Closed theorem form of spim 1642. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.) |
Theorem | spimh 1641 | Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1642 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 1642 | Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1642 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 1643 | 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 1644 | Deduction version of spime 1645. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) |
Theorem | spime 1645 | 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 1646 | 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 1647 | 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 1648 | 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 1649 | 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 1650 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | cbv2 1651 | 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 1652 | 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 1653 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Theorem | cbvexh 1654 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.) |
Theorem | cbvex 1655 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | chvar 1656 | Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Theorem | equvini 1657 | 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 1658 | A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1657.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.) |
Theorem | nfald 1659 | 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 1660 | 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 1661 | 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 1662 |
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 1674.
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 1737, sbcom2 1879 and sbid2v 1888). Note that our definition is valid even when and are replaced with the same variable, as sbid 1673 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 1883 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 1886. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1783 and sb6 1782. 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 1663 | Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
Theorem | sbbii 1664 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb1 1665 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb2 1666 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ1 1667 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ2 1668 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | stdpc7 1669 | One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1607.) 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 1670 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ12r 1671 | An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | sbequ12a 1672 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid 1673 | 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 1674 | 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 1675 | 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 1676 | 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 1677 | Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.) |
Theorem | sb6x 1678 | Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | nfs1f 1679 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbs1f 1680 | 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 1681 | Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.) |
Theorem | sbequ6 1682 | Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.) |
Theorem | sbt 1683 | A substitution into a theorem remains true. (See chvar 1656 and chvarv 1828 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equsb1 1684 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | equsb2 1685 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbiedh 1686 | Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1689). New proofs should use sbied 1687 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
Theorem | sbied 1687 | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1690). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sbiedv 1688* | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1690). (Contributed by NM, 7-Jan-2017.) |
Theorem | sbieh 1689 | Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1690 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.) |
Theorem | sbie 1690 | 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 1691 | A property related to substitution that unlike equs5 1726 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | equs5e 1692 | A property related to substitution that unlike equs5 1726 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.) |
Theorem | ax11e 1693 | Analogue to ax-11 1413 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.) |
Theorem | ax10oe 1694 | Quantifier Substitution for existential quantifiers. Analogue to ax10o 1619 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.) |
Theorem | drex1 1695 | 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 1696 | 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 1697 | 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 1698 | A version of sb4 1729 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | equs45f 1699 | Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.) |
Theorem | sb6f 1700 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.) |
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