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Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2reuswap 3001* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)

Theoremreuind 3002* Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.)

Theorem2rmorex 3003* Double restricted quantification with "at most one," analogous to 2moex 2247. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theorem2reu5lem1 3004* Lemma for 2reu5 3007. Note that does not mean "there is exactly one in and exactly one in such that holds;" see comment for 2eu5 2260. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theorem2reu5lem2 3005* Lemma for 2reu5 3007. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theorem2reu5lem3 3006* Lemma for 2reu5 3007. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3110. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theorem2reu5 3007* Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2260 and reu3 2989. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

2.1.7  Conditional equality (experimental)

This is a very useless definition, which "abbreviates" as CondEq . What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific 3-ary operation .

This is all used as part of a metatheorem: we want to say that and are provable, for any expressions or in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations.

The metatheorem comes with a disjoint variables assumption: every variable in is assumed disjoint from except itself. For such a proof by induction, we must consider each of the possible forms of . If it is a variable other than , then we have CondEq or CondEq , which is provable by cdeqth 3012 and reflexivity. Since we are only working with class and wff expressions, it can't be itself in set.mm, but if it was we'd have to also prove CondEq (where set equality is being used on the right).

Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each set variable parameter to the operation, we must consider if it is equal to or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder).

In each of the primitive proofs, we are allowed to assume that is disjoint from and vice versa, because this is maintained through the induction. This is how we satisfy the DV assumptions of cdeqab1 3017 and cdeqab 3015.

Syntaxwcdeq 3008 Extend wff notation to include conditional equality. This is a technical device used in the proof that is the not-free predicate, and that definitions are conservative as a result.
CondEq

Definitiondf-cdeq 3009 Define conditional equality. All the notation to the left of the is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqi 3010 Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqri 3011 Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqth 3012 Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqnot 3013 Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqal 3014* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqab 3015* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqal1 3016* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqab1 3017* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqim 3018 Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq        CondEq

Theoremcdeqcv 3019 Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqeq 3020 Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq        CondEq

Theoremcdeqel 3021 Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq        CondEq

Theoremnfcdeq 3022* If we have a conditional equality proof, where is and is , and in fact does not have free in it according to , then unconditionally. This proves that is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremnfccdeq 3023* Variation of nfcdeq 3022 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremru 3024 Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.

In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as , asserted that any collection of sets is a set i.e. belongs to the universe of all sets. In particular, by substituting (the "Russell class") for , it asserted , meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove . This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system.

In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 4195 asserting that is a set only when it is smaller than some other set . However, Zermelo was then faced with a "chicken and egg" problem of how to show is a set, leading him to introduce the set-building axioms of Null Set 0ex 4187, Pairing prex 4254, Union uniex 4553, Power Set pwex 4230, and Infinity omex 7389 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 5367 (whose modern formalization is due to Skolem, also in 1922). Thus, in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics!

Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than set variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287).

Russell himself continued in a different direction, avoiding the paradox with his "theory of types." Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contradicting ZF and NBG set theories, and it has other bizarre consequences: when sets become too huge (beyond the size of those used in standard mathematics), the Axiom of Choice ac4 8147 and Cantor's Theorem canth 6336 are provably false! (See ncanth 6337 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 4178 replaces ax-rep 4168) with ax-sep 4178 restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic," J. Symb. Logic 9:1-19 (1944).

Under our ZF set theory, every set is a member of the Russell class by elirrv 7356 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7359). See ruALT 7360 for an alternate proof of ru 3024 derived from that fact. (Contributed by NM, 7-Aug-1994.)

2.1.9  Proper substitution of classes for sets

Syntaxwsbc 3025 Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class for set variable in wff ."

Definitiondf-sbc 3026 Define the proper substitution of a class for a set.

When is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3051 for our definition, which always evaluates to true for proper classes.

Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 3027 below). For example, if is a proper class, Quine's substitution of for in evaluates to rather than our falsehood. (This can be seen by substituting , , and for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactical breakdown of , and it does not seem possible to express it with a single closed formula.

If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 3027, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3026 in the form of sbc8g 3032. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of in every use of this definition) we allow direct reference to df-sbc 3026 and assert that is always false when is a proper class.

The theorem sbc2or 3033 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3027.

The related definition df-csb 3116 defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.)

Theoremdfsbcq 3027 This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 3026 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 3028 instead of df-sbc 3026. (dfsbcq2 3028 is needed because unlike Quine we do not overload the df-sb 1640 syntax.) As a consequence of these theorems, we can derive sbc8g 3032, which is a weaker version of df-sbc 3026 that leaves substitution undefined when is a proper class.

However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3032, so we will allow direct use of df-sbc 3026 after theorem sbc2or 3033 below. Proper substiution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.)

Theoremdfsbcq2 3028 This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1640 and substitution for class variables df-sbc 3026. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3027. (Contributed by NM, 31-Dec-2016.)

Theoremsbsbc 3029 Show that df-sb 1640 and df-sbc 3026 are equivalent when the class term in df-sbc 3026 is a set variable. This theorem lets us reuse theorems based on df-sb 1640 for proofs involving df-sbc 3026. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)

Theoremsbceq1d 3030 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremsbceq1dd 3031 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremsbc8g 3032 This is the closest we can get to df-sbc 3026 if we start from dfsbcq 3027 (see its comments) and dfsbcq2 3028. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)

Theoremsbc2or 3033* The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for behavior at proper classes, matching the sbc5 3049 (false) and sbc6 3051 (true) conclusions. This is interesting since dfsbcq 3027 and dfsbcq2 3028 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem doesn't tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable that or may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.)

Theoremsbcex 3034 By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.)

Theoremsbceq1a 3035 Equality theorem for class substitution. Class version of sbequ12 1891. (Contributed by NM, 26-Sep-2003.)

Theoremsbceq2a 3036 Equality theorem for class substitution. Class version of sbequ12r 1892. (Contributed by NM, 4-Jan-2017.)

Theoremspsbc 3037 Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1996 and rspsbc 3103. (Contributed by NM, 16-Jan-2004.)

Theoremspsbcd 3038 Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1996 and rspsbc 3103. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremsbcth 3039 A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.)

Theoremsbcthdv 3040* Deduction version of sbcth 3039. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremsbcid 3041 An identity theorem for substitution. See sbid 1894. (Contributed by Mario Carneiro, 18-Feb-2017.)

Theoremnfsbc1d 3042 Deduction version of nfsbc1 3043. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfsbc1 3043 Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfsbc1v 3044* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfsbcd 3045 Deduction version of nfsbc 3046. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfsbc 3046 Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremsbcco 3047* A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremsbcco2 3048* A composition law for class substitution. Importantly, may occur free in the class expression substituted for . (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremsbc5 3049* An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremsbc6g 3050* An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremsbc6 3051* An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)

Theoremsbc7 3052* An equivalence for class substitution in the spirit of df-clab 2303. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcbvsbc 3053 Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremcbvsbcv 3054* Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremsbciegft 3055* Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3056.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremsbciegf 3056* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremsbcieg 3057* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)

Theoremsbcie2g 3058* Conversion of implicit substitution to explicit class substitution. This version of sbcie 3059 avoids a disjointness condition on by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremsbcie 3059* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)

Theoremsbciedf 3060* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)

Theoremsbcied 3061* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Theoremsbcied2 3062* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Theoremelrabsf 3063 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2956 has implicit substitution). The hypothesis specifies that must not be a free variable in . (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremeqsbc3 3064* Substitution applied to an atomic wff. Set theory version of eqsb3 2417. (Contributed by Andrew Salmon, 29-Jun-2011.)

Theoremsbcng 3065 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)

Theoremsbcimg 3066 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)

Theoremsbcan 3067 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)

Theoremsbcang 3068 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)

Theoremsbcor 3069 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)

Theoremsbcorg 3070 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)

Theoremsbcbig 3071 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremsbcal 3072* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)

Theoremsbcalg 3073* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)

Theoremsbcex2 3074* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)

Theoremsbcexg 3075* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)

Theoremsbceqal 3076* Set theory version of sbeqal1 26745. (Contributed by Andrew Salmon, 28-Jun-2011.)

Theoremsbeqalb 3077* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)

Theoremsbcbid 3078 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

Theoremsbcbidv 3079* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

Theoremsbcbii 3080 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)

TheoremsbcbiiOLD 3081 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) (New usage is discouraged.)

Theoremeqsbc3r 3082* eqsbc3 3064 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 28127 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremsbc3ang 3083 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcel1gv 3084* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcel2gv 3085* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcimdv 3086* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)

Theoremsbctt 3087 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremsbcgf 3088 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbc19.21g 3089 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

Theoremsbcg 3090* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3088. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremsbc2iegf 3091* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theoremsbc2ie 3092* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremsbc2iedv 3093* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Theoremsbc3ie 3094* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremsbccomlem 3095* Lemma for sbccom 3096. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)

Theoremsbccom 3096* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Theoremsbcralt 3097* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)

Theoremsbcrext 3098* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremsbcralg 3099* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcrexg 3100* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

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