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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 19.12 1601 | Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.19 1602 | Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
Theorem | 19.21-2 1603 | Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.) |
Theorem | nf2 1604 | An alternate definition of df-nf 1396, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | nf3 1605 | An alternate definition of df-nf 1396. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | nf4dc 1606 | Variable is effectively not free in iff is always true or always false, given a decidability condition. The reverse direction, nf4r 1607, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.) |
DECID | ||
Theorem | nf4r 1607 | If is always true or always false, then variable is effectively not free in . The converse holds given a decidability condition, as seen at nf4dc 1606. (Contributed by Jim Kingdon, 21-Jul-2018.) |
Theorem | 19.36i 1608 | Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
Theorem | 19.36-1 1609 | Closed form of 19.36i 1608. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.) |
Theorem | 19.37-1 1610 | One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.) |
Theorem | 19.37aiv 1611* | Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.38 1612 | Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.23t 1613 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
Theorem | 19.23 1614 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
Theorem | 19.32dc 1615 | Theorem 19.32 of [Margaris] p. 90, where is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.) |
DECID | ||
Theorem | 19.32r 1616 | One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if is decidable, as seen at 19.32dc 1615. (Contributed by Jim Kingdon, 28-Jul-2018.) |
Theorem | 19.31r 1617 | One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.) |
Theorem | 19.44 1618 | Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
Theorem | 19.45 1619 | Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
Theorem | 19.34 1620 | Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.41h 1621 | Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1622 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
Theorem | 19.41 1622 | Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) |
Theorem | 19.42h 1623 | Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1624 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.) |
Theorem | 19.42 1624 | Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |
Theorem | excom13 1625 | Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.) |
Theorem | exrot3 1626 | Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.) |
Theorem | exrot4 1627 | Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.) |
Theorem | nexr 1628 | Inference from 19.8a 1528. (Contributed by Jeff Hankins, 26-Jul-2009.) |
Theorem | exan 1629 | Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | hbexd 1630 | Deduction form of bound-variable hypothesis builder hbex 1573. (Contributed by NM, 2-Jan-2002.) |
Theorem | eeor 1631 | Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.) |
Theorem | a9e 1632 | 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 1382 through ax-14 1451 and ax-17 1465, all axioms other than ax-9 1470 are believed to be theorems of free logic, although the system without ax-9 1470 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | a9ev 1633* | At least one individual exists. Weaker version of a9e 1632. (Contributed by NM, 3-Aug-2017.) |
Theorem | ax9o 1634 | An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | equid 1635 |
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 1636 | 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 1637 | One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1701.) 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 1638 | Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.) |
Theorem | ax6evr 1639* | A commuted form of a9ev 1633. The naming reflects how axioms were numbered in the Metamath Proof Explorer as of 2020 (a numbering which we eventually plan to adopt here too, but until this happens everywhere only some theorems will have it). (Contributed by BJ, 7-Dec-2020.) |
Theorem | equcom 1640 | Commutative law for equality. (Contributed by NM, 20-Aug-1993.) |
Theorem | equcomd 1641 | Deduction form of equcom 1640, symmetry of equality. For the versions for classes, see eqcom 2091 and eqcomd 2094. (Contributed by BJ, 6-Oct-2019.) |
Theorem | equcoms 1642 | An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.) |
Theorem | equtr 1643 | A transitive law for equality. (Contributed by NM, 23-Aug-1993.) |
Theorem | equtrr 1644 | A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.) |
Theorem | equtr2 1645 | A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equequ1 1646 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) |
Theorem | equequ2 1647 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) |
Theorem | elequ1 1648 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
Theorem | elequ2 1649 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11i 1650 | Inference that has ax-11 1443 (without ) as its conclusion and doesn't require ax-10 1442, ax-11 1443, or ax-12 1448 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 1651 |
Show that ax-10o 1652 can be derived from ax-10 1442. An open problem is
whether this theorem can be derived from ax-10 1442 and the others when
ax-11 1443 is replaced with ax-11o 1752. See theorem ax10 1653
for the
rederivation of ax-10 1442 from ax10o 1651.
Normally, ax10o 1651 should be used rather than ax-10o 1652, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) |
Axiom | ax-10o 1652 |
Axiom ax-10o 1652 ("o" for "old") was the
original version of ax-10 1442,
before it was discovered (in May 2008) that the shorter ax-10 1442 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 1651. Normally, ax10o 1651 should be used rather than ax-10o 1652, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | ax10 1653 |
Rederivation of ax-10 1442 from original version ax-10o 1652. See theorem
ax10o 1651 for the derivation of ax-10o 1652 from ax-10 1442.
This theorem should not be referenced in any proof. Instead, use ax-10 1442 above so that uses of ax-10 1442 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.) |
Theorem | hbae 1654 | All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | nfae 1655 | All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbaes 1656 | Rule that applies hbae 1654 to antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | hbnae 1657 | 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 1658 | All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbnaes 1659 | Rule that applies hbnae 1657 to antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | naecoms 1660 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) |
Theorem | equs4 1661 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) |
Theorem | equsalh 1662 | A useful equivalence related to substitution. New proofs should use equsal 1663 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Theorem | equsal 1663 | 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 1664 | A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | equsexd 1665 | Deduction form of equsex 1664. (Contributed by Jim Kingdon, 29-Dec-2017.) |
Theorem | dral1 1666 | 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 1667 | 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 1668 | 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 1669 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | drnf2 1670 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | spimth 1671 | Closed theorem form of spim 1674. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.) |
Theorem | spimt 1672 | Closed theorem form of spim 1674. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.) |
Theorem | spimh 1673 | Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1674 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 1674 | Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1674 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 1675 | 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 1676 | Deduction version of spime 1677. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) |
Theorem | spime 1677 | 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 1678 | 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 1679 | 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 1680 | 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 1681 | 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 1682 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | cbv2 1683 | 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 1684 | 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 1685 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Theorem | cbvexh 1686 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.) |
Theorem | cbvex 1687 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | chvar 1688 | Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Theorem | equvini 1689 | 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 1690 | A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1689.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.) |
Theorem | nfald 1691 | 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 1692 | 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 1693 | 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 1694 |
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 1706.
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 1769, sbcom2 1912 and sbid2v 1921). Note that our definition is valid even when and are replaced with the same variable, as sbid 1705 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 1916 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 1919. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1816 and sb6 1815. 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 1695 | Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
Theorem | sbbii 1696 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb1 1697 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb2 1698 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ1 1699 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ2 1700 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
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