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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | alim 1401 | Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.) |
Theorem | al2imi 1402 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alanimi 1403 | Variant of al2imi 1402 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
Syntax | wnf 1404 | Extend wff definition to include the not-free predicate. |
Definition | df-nf 1405 |
Define the not-free predicate for wffs. This is read " is not free
in ".
Not-free means that the value of cannot affect the
value of ,
e.g., any occurrence of in
is effectively
bound by a "for all" or something that expands to one (such as
"there
exists"). In particular, substitution for a variable not free in a
wff
does not affect its value (sbf 1718). An example of where this is used is
stdpc5 1531. See nf2 1614 for an alternate definition which
does not involve
nested quantifiers on the same variable.
Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition. To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, is effectively not free in the bare expression , even though would be considered free in the usual textbook definition, because the value of in the expression cannot affect the truth of the expression (and thus substitution will not change the result). (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfi 1406 | Deduce that is not free in from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbth 1407 |
No variable is (effectively) free in a theorem.
This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form from smaller formulas of this form. These are useful for constructing hypotheses that state " is (effectively) not free in ." (Contributed by NM, 5-Aug-1993.) |
Theorem | nfth 1408 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfnth 1409 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
Theorem | nftru 1410 | The true constant has no free variables. (This can also be proven in one step with nfv 1476, but this proof does not use ax-17 1474.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
Theorem | alimdh 1411 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.) |
Theorem | albi 1412 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimih 1413 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | albii 1414 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
Theorem | 2albii 1415 | Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
Theorem | hbxfrbi 1416 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | nfbii 1417 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfr 1418 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfrd 1419 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | alcoms 1420 | Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.) |
Theorem | hbal 1421 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | alcom 1422 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimdh 1423 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | albidh 1424 | Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.26 1425 | Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
Theorem | 19.26-2 1426 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |
Theorem | 19.26-3an 1427 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |
Theorem | 19.33 1428 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrot3 1429 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
Theorem | alrot4 1430 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.) |
Theorem | albiim 1431 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
Theorem | 2albiim 1432 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |
Theorem | hband 1433 | Deduction form of bound-variable hypothesis builder hban 1494. (Contributed by NM, 2-Jan-2002.) |
Theorem | hb3and 1434 | Deduction form of bound-variable hypothesis builder hb3an 1497. (Contributed by NM, 17-Feb-2013.) |
Theorem | hbald 1435 | Deduction form of bound-variable hypothesis builder hbal 1421. (Contributed by NM, 2-Jan-2002.) |
Syntax | wex 1436 | Extend wff definition to include the existential quantifier ("there exists"). |
Axiom | ax-ie1 1437 | is bound in . One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Axiom | ax-ie2 1438 | Define existential quantification. means "there exists at least one set such that is true." One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Theorem | hbe1 1439 | is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | nfe1 1440 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | 19.23ht 1441 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.) |
Theorem | 19.23h 1442 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.) |
Theorem | alnex 1443 | 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 dfexdc 1445 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 1444 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |
Theorem | dfexdc 1445 | Defining given decidability. It is common in classical logic to define as but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1446. (Contributed by Jim Kingdon, 15-Mar-2018.) |
DECID | ||
Theorem | exalim 1446 | One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1445. (Contributed by Jim Kingdon, 29-Jul-2018.) |
The equality predicate was introduced above in wceq 1299 for use by df-tru 1302. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Theorem | weq 1447 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1447 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 1299. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1447 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1299. 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 1448 |
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 1449 of predicate calculus in terms of the wceq 1299 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 1449 |
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 1449 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 1448. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1449 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1448. 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 1450 |
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 1653). Axiom scheme C8'
in [Megill] p. 448 (p. 16 of the preprint).
Also appears as Axiom C7 of
[Monk2] p. 105.
Axioms ax-8 1450 through ax-16 1753 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 1753 and ax-17 1474 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 1753 and ax-17 1474 only. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-10 1451 |
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 1662 ("o" for "old") and was replaced with this shorter ax-10 1451 in May 2008. The old axiom is proved from this one as theorem ax10o 1661. Conversely, this axiom is proved from ax-10o 1662 as theorem ax10 1663. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-11 1452 |
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 1825).
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 1766, ax11v2 1759 and ax-11o 1762. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-i12 1453 |
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 1457 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Axiom | ax-bndl 1454 |
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 implies ax-i12 1453 as can be seen at axi12 1462. Whether ax-bndl 1454 can be proved from the remaining axioms including ax-i12 1453 is not known. 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 1455 |
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 1393. Conditional forms of the converse are given by ax-12 1457, ax-16 1753, and ax-17 1474. 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 1716. (Contributed by NM, 5-Aug-1993.) |
Theorem | sp 1456 | Specialization. Another name for ax-4 1455. (Contributed by NM, 21-May-2008.) |
Theorem | ax-12 1457 | Rederive the original version of the axiom from ax-i12 1453. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ax12or 1458 | Another name for ax-i12 1453. (Contributed by NM, 3-Feb-2015.) |
Axiom | ax-13 1459 | 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 1460 | 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 1461 | 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 1391, ax-8 1450, ax-12 1457, and ax-gen 1393. This shows that this can be proved without ax-9 1479, even though the theorem equid 1645 cannot be. A shorter proof using ax-9 1479 is obtainable from equid 1645 and hbth 1407.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) |
Theorem | axi12 1462 | Proof that ax-i12 1453 follows from ax-bndl 1454. So that we can track which theorems rely on ax-bndl 1454, proofs should reference ax-i12 1453 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.) |
Theorem | alequcom 1463 | 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 1464 | A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.) |
Theorem | nalequcoms 1465 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.) |
Theorem | nfr 1466 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |
Theorem | nfri 1467 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfrd 1468 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | alimd 1469 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | alrimi 1470 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | nfd 1471 | Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | nfdh 1472 | Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | nfrimi 1473 | Moving an antecedent outside . (Contributed by Jim Kingdon, 23-Mar-2018.) |
Axiom | ax-17 1474* |
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 1475* | ax-17 1474 with antecedent. (Contributed by NM, 1-Mar-2013.) |
Theorem | nfv 1476* | If is not present in , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfvd 1477* | nfv 1476 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1532. (Contributed by Mario Carneiro, 6-Oct-2016.) |
Axiom | ax-i9 1478 | 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 1455 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 1642, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Theorem | ax-9 1479 | Derive ax-9 1479 from ax-i9 1478, the modified version for intuitionistic logic. Although ax-9 1479 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1478. (Contributed by NM, 3-Feb-2015.) |
Theorem | equidqe 1480 | equid 1645 with some quantification and negation without using ax-4 1455 or ax-17 1474. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) |
Theorem | ax4sp1 1481 | A special case of ax-4 1455 without using ax-4 1455 or ax-17 1474. (Contributed by NM, 13-Jan-2011.) |
Axiom | ax-ial 1482 | is not free in . One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Axiom | ax-i5r 1483 | Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Theorem | spi 1484 | Inference reversing generalization (specialization). (Contributed by NM, 5-Aug-1993.) |
Theorem | sps 1485 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | spsd 1486 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |
Theorem | nfbidf 1487 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | hba1 1488 | 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 1489 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | a5i 1490 | Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfnf1 1491 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbim 1492 | 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 1493 | 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 1494 | 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 1495 | If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | hb3or 1496 | If is not free in , , and , it is not free in . (Contributed by NM, 14-Sep-2003.) |
Theorem | hb3an 1497 | If is not free in , , and , it is not free in . (Contributed by NM, 14-Sep-2003.) |
Theorem | hba2 1498 | Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
Theorem | hbia1 1499 | Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
Theorem | 19.3h 1500 | 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.) |
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