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Type | Label | Description |
---|---|---|

Statement | ||

Theorem | 19.26OLD 1301 | Obsolete proof of 19.26 1300 as of 4-Jul-2014. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.26-2 1302 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |

Theorem | 19.26-3an 1303 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |

Theorem | 19.33 1304 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | alrot3 1305 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |

Theorem | alrot4 1306 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.) |

Theorem | alrot4OLD 1307 | Obsolete proof of alrot4 1306 as of 28-Jun-2014. (Contributed by NM, 2-Feb-2005.) |

Theorem | albiim 1308 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |

Theorem | 2albiim 1309 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |

Theorem | hband 1310 | Deduction form of bound-variable hypothesis builder hban 1371. (Contributed by NM, 2-Jan-2002.) |

Theorem | hb3and 1311 | Deduction form of bound-variable hypothesis builder hb3an 1374. (Contributed by NM, 17-Feb-2013.) |

Theorem | hbald 1312 | Deduction form of bound-variable hypothesis builder hbal 1296. (Contributed by NM, 2-Jan-2002.) |

Syntax | wex 1313 | Extend wff definition to include the existential quantifier ("there exists"). |

Axiom | ax-ie1 1314 | is bound in . Axiom 9 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |

Axiom | ax-ie2 1315 | Define existential quantification. means "there exists at least one set such that is true." Axiom 10 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |

Theorem | hbe1 1316 | is not free in . (Contributed by NM, 5-Aug-1993.) |

Theorem | nfe1 1317 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | 19.23t 1318 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.) |

Theorem | 19.23 1319 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.) |

Theorem | alnex 1320 | Theorem 19.7 of [Margaris] p. 89. To read this intuitionistically, think of it as "if can be refuted for all , then it is not possible to find an for which holds" (and likewise for the converse). Comparing this with df-ex 2063 illustrates that statements which look similar (to someone used to classical logic) can be different intuitionistically due to different placement of negations. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 1-Feb-2015.) (Revised by Mario Carneiro, 12-May-2015.) |

Theorem | nex 1321 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |

Theorem | dfexdc 1322 | Defining given decidability. It is common in classical logic to define as but in intuitionistic logic, that definition only holds under certain conditions. (Contributed by Jim Kingdon, 15-Mar-2018.) |

DECID | ||

1.3.2 Introduce equality axioms | ||

Syntax | cv 1323 |
This syntax construction states that a variable , which has been
declared to be a set variable by $f statement vx, is also a class
expression. See comments in set.mm for justification.
While it is tempting and perhaps occasionally useful to view cv 1323 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1323 is intrinsically no different from any other class-building syntax. (The description above applies to set theory, not predicate calculus. The purpose of introducing here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1325 of predicate calculus from the wceq 1324 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.) |

Syntax | wceq 1324 |
Extend wff definition to include class equality.
(The purpose of introducing here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1325 of predicate calculus in terms of the wceq 1324 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the in could be the of either weq 1325 or wceq 1324, although mathematically it makes no difference. The class variables and are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus.) |

Theorem | weq 1325 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1325 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 1324. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1325 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1324. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.) |

Syntax | wcel 1326 |
Extend wff definition to include the membership connective between
classes.
(The purpose of introducing here is to allow us to express i.e. "prove" the wel 1327 of predicate calculus in terms of the wceq 1324 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables and are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus.) |

Theorem | wel 1327 |
Extend wff definition to include atomic formulas with the epsilon
(membership) predicate. This is read " is an element of
," " is a member of ," " belongs to ,"
or " contains
." Note: The
phrase " includes
" means
" is a subset of
;" to use it also
for
, as some authors occasionally do, is poor form
and causes
confusion, according to George Boolos (1992 lecture at MIT).
This syntactical construction introduces a binary non-logical predicate symbol (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. (Instead of introducing wel 1327 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 1326. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1327 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1326. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.) |

Axiom | ax-8 1328 |
Axiom of Equality. One of the equality and substitution axioms of
predicate calculus with equality. This is similar to, but not quite, a
transitive law for equality (proved later as equtr 1485). Axiom scheme C8'
in [Megill] p. 448 (p. 16 of the preprint).
Also appears as Axiom C7 of
[Monk2] p. 105.
Axioms ax-8 1328 through ax-16 1581 are the axioms having to do with equality, substitution, and logical properties of our binary predicate (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1581 and ax-17 1350 are still valid even when , , and are replaced with the same variable because they do not have any distinct variable (Metamath's $d) restrictions. Distinct variable restrictions are required for ax-16 1581 and ax-17 1350 only. (Contributed by NM, 5-Aug-1993.) |

Axiom | ax-10 1329 |
Axiom of Quantifier Substitution. One of the equality and substitution
axioms of predicate calculus with equality. Appears as Lemma L12 in
[Megill] p. 445 (p. 12 of the preprint).
The original version of this axiom was ax-10o 1494 ("o" for "old") and was replaced with this shorter ax-10 1329 in May 2008. The old axiom is proved from this one as theorem ax10o 1493. Conversely, this axiom is proved from ax-10o 1494 as theorem ax10 1495. (Contributed by NM, 5-Aug-1993.) |

Axiom | ax-11 1330 |
Axiom of Variable Substitution. One of the 5 equality axioms of predicate
calculus. The final consequent is a way of
expressing "
substituted for in
wff " (cf.
sb6 1648). 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.
Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1594, ax11v2 1587 and ax-11o 1590. (Contributed by NM, 5-Aug-1993.) |

Axiom | ax-i12 1331 |
Axiom of Quantifier Introduction. One of the equality and substitution
axioms of predicate calculus with equality. Informally, it says that
whenever is
distinct from and
, and
is true,
then quantified with is also true. In other words,
is irrelevant to the truth of
. Axiom scheme C9' 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.
This axiom has been modified from the original ax-12 1335 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |

Axiom | ax-bnd 1332 |
Axiom of bundling. The general idea of this axiom is that two variables
are either distinct or non-distinct. That idea could be expressed as
. However, we instead choose an axiom
which has many of the same consequences, but which is different with
respect to a universe which contains only one object.
holds
if and are the same variable,
likewise for and ,
and holds if
is distinct from
the others (and the universe has at least two objects).
As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability). This axiom is similar to ax-i12 1331, but appears to be stronger. At least for now, we keep them both as distinct axioms, but they serve similar purposes. The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.) |

Axiom | ax-4 1333 |
Axiom of Specialization. A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all , it is true for any
specific (that
would typically occur as a free variable in the wff
substituted for ). (A free variable is one that does not occur in
the scope of a quantifier: and are
both free in ,
but only is free
in .) Axiom
scheme C5' in [Megill]
p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski]
p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1269. Conditional forms of the converse are given by ax-12 1335, ax-16 1581, and ax-17 1350. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1545. (Contributed by NM, 5-Aug-1993.) |

Theorem | sp 1334 | Specialization. Another name for ax-4 1333. (Contributed by NM, 21-May-2008.) |

Theorem | ax-12 1335 | Rederive the original version of the axiom from ax-i12 1331. (Contributed by Mario Carneiro, 3-Feb-2015.) |

Theorem | ax12or 1336 | Another name for ax-i12 1331. (Contributed by NM, 3-Feb-2015.) |

Axiom | ax-13 1337 | Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.) |

Axiom | ax-14 1338 | Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.) |

Theorem | hbequid 1339 | 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. (The proof uses only ax-5 1267, ax-8 1328, ax-12 1335, and ax-gen 1269. This shows that this can be proved without ax-9 1355, even though the theorem equid 1480 cannot be. A shorter proof using ax-9 1355 is obtainable from equid 1480 and hbth 1283.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) |

Theorem | alequcom 1340 | Commutation law for identical variable specifiers. The antecedent and consequent are true when and are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.) |

Theorem | alequcoms 1341 | A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.) |

Theorem | nalequcoms 1342 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.) |

Theorem | nfr 1343 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |

Theorem | nfri 1344 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nfrd 1345 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | alrimi 1346 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nfd 1347 | Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nfdh 1348 | Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | nfrimi 1349 | Moving an antecedent outside . (Contributed by Jim Kingdon, 23-Mar-2018.) |

1.3.3 Axiom ax-17 - first use of the $d distinct
variable statement | ||

Axiom | ax-17 1350* |
Axiom to quantify a variable over a formula in which it does not occur.
Axiom C5 in [Megill] p. 444 (p. 11 of the
preprint). Also appears as
Axiom B6 (p. 75) of system S2 of [Tarski]
p. 77 and Axiom C5-1 of
[Monk2] p. 113.
(Contributed by NM, 5-Aug-1993.) |

Theorem | a17d 1351* | ax-17 1350 with antecedent. (Contributed by NM, 1-Mar-2013.) |

Theorem | nfv 1352* | If is not present in , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nfvd 1353* | nfv 1352 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1403. (Contributed by Mario Carneiro, 6-Oct-2016.) |

1.3.4 Introduce Axiom of Existence | ||

Axiom | ax-i9 1354 | Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. One thing this axiom tells us is that at least one thing exists (although ax-4 1333 and possibly others also tell us that, i.e. they are not valid in the empty domain of a "free logic"). In this form (not requiring that and be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) Another name for this theorem is a9e 1478, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.) |

Theorem | ax-9 1355 | Derive ax-9 1355 from ax-i9 1354, the modified version for intuitionistic logic. Although ax-9 1355 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1354. (Contributed by NM, 3-Feb-2015.) |

Theorem | equidqe 1356 | equid 1480 with some quantification and negation without using ax-4 1333 or ax-17 1350. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) |

Theorem | equidqeOLD 1357 | Obsolete proof of equidqe 1356 as of 27-Feb-2014. (Contributed by NM, 13-Jan-2011.) |

Theorem | ax4sp1 1358 | A special case of ax-4 1333 without using ax-4 1333 or ax-17 1350. (Contributed by NM, 13-Jan-2011.) |

1.3.5 Additional intuitionistic
axioms | ||

Axiom | ax-ial 1359 | is not free in . Axiom 7 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |

Axiom | ax-i5r 1360 | Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.) |

1.3.6 Predicate calculus including ax-4, without
distinct variables | ||

Theorem | a4i 1361 | Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.) |

Theorem | a4s 1362 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |

Theorem | a4sd 1363 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |

Theorem | nfbidf 1364 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |

Theorem | hba1 1365 | is not free in . Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.) |

Theorem | nfa1 1366 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | a5i 1367 | Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.) |

Theorem | nfnf1 1368 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | hbim 1369 | If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.) |

Theorem | hbor 1370 | If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |

Theorem | hban 1371 | If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.) |

Theorem | hbbi 1372 | If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.) |

Theorem | hb3or 1373 | If is not free in , , and , it is not free in . (Contributed by NM, 14-Sep-2003.) |

Theorem | hb3an 1374 | If is not free in , , and , it is not free in . (Contributed by NM, 14-Sep-2003.) |

Theorem | hba2 1375 | Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |

Theorem | hbia1 1376 | Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |

Theorem | 19.3 1377 | A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-May-2007.) |

Theorem | 19.16 1378 | Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.17 1379 | Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.21 1380 | Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " is not free in ." (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.21-2 1381 | Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.) |

Theorem | stdpc5 1382 | An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis can be thought of as emulating " is not free in ." With this convention, the meaning of "not free" is less restrictive than the usual textbook definition; for example would not (for us) be free in by hbequid 1339. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) |

Theorem | 19.21bi 1383 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.21bbi 1384 | Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.) |

Theorem | 19.27 1385 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.28 1386 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | nfan1 1387 | A closed form of nfan 1388. (Contributed by Mario Carneiro, 3-Oct-2016.) |

Theorem | nfan 1388 | If is not free in and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) |

Theorem | nf3an 1389 | If is not free in , , and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nfand 1390 | If in a context is not free in and , it is not free in . (Contributed by Mario Carneiro, 7-Oct-2016.) |

Theorem | nf3and 1391 | Deduction form of bound-variable hypothesis builder nf3an 1389. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.) |

Theorem | hbim1 1392 | A closed form of hbim 1369. (Contributed by NM, 5-Aug-1993.) |

Theorem | nfim1 1393 | A closed form of nfim 1394. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |

Theorem | nfim 1394 | If is not free in and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |

Theorem | hbimd 1395 | Deduction form of bound-variable hypothesis builder hbim 1369. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.) |

Theorem | nfor 1396 | If is not free in and , it is not free in . (Contributed by Jim Kingdon, 11-Mar-2018.) |

Theorem | hbbid 1397 | Deduction form of bound-variable hypothesis builder hbbi 1372. (Contributed by NM, 1-Jan-2002.) |

Theorem | nfal 1398 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nfnf 1399 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |

Theorem | nfalt 1400 | Closed form of nfal 1398. (Contributed by Jim Kingdon, 11-May-2018.) |

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