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Theorem List for Metamath Proof Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsb5f 2501 Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 2478 and does not require the nonfreeness hypothesis. Theorem sb5 2272 replaces the nonfreeness hypothesis with a disjoint variable condition on 𝑥, 𝑦 and requires fewer axioms. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
𝑦𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremnfsb4t 2502 A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (closed form of nfsb4 2503). Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.)
(∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
 
Theoremnfsb4 2503 A variable not free in a proposition remains so after substitution in that proposition with a distinct variable. Usage of this theorem is discouraged because it depends on ax-13 2371. Theorem nfsb 2526 replaces the distinctor with a disjoint variable condition. Visit also nfsbv 2329 for a weaker version of nfsb 2526 not requiring ax-13 2371. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
𝑧𝜑       (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
 
Theoremsbequ8 2504 Elimination of equality from antecedent after substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.) Revise df-sb 2071. (Revised by Wolf Lammen, 28-Jul-2023.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))
 
Theoremsbie 2505 Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev 2313 and sbievw 2099. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)
 
Theoremsbied 2506 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2505) Usage of this theorem is discouraged because it depends on ax-13 2371. See sbiedw 2314, sbiedvw 2100 for variants using disjoint variables, but requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 
Theoremsbiedv 2507* Conversion of implicit substitution to explicit substitution (deduction version of sbie 2505). Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker sbiedvw 2100 when possible. (Contributed by NM, 7-Jan-2017.) (New usage is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 
Theorem2sbiev 2508* Conversion of double implicit substitution to explicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. See 2sbievw 2101 for a version with extra disjoint variables, but based on fewer axioms. (Contributed by AV, 29-Jul-2023.) (New usage is discouraged.)
((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))       ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)
 
Theoremsbcom3 2509 Substituting 𝑦 for 𝑥 and then 𝑧 for 𝑦 is equivalent to substituting 𝑧 for both 𝑥 and 𝑦. Usage of this theorem is discouraged because it depends on ax-13 2371. For a version requiring a disjoint variable, but fewer axioms, see sbcom3vv 2102. (Contributed by Giovanni Mascellani, 8-Apr-2018.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 16-Sep-2018.) (Proof shortened by Wolf Lammen, 16-Sep-2018.) (New usage is discouraged.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)
 
Theoremsbco 2510 A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. See sbcov 2254 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (New usage is discouraged.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 
Theoremsbid2 2511 An identity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Check out sbid2vw 2256 for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
𝑥𝜑       ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)
 
Theoremsbid2v 2512* An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). Usage of this theorem is discouraged because it depends on ax-13 2371. See sbid2vw 2256 for a version with an extra disjoint variable condition requiring fewer axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)
 
Theoremsbidm 2513 An idempotent law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (New usage is discouraged.)
([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 
Theoremsbco2 2514 A composition law for substitution. For versions requiring fewer axioms, but more disjoint variable conditions, see sbco2v 2332 and sbco2vv 2104. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) (New usage is discouraged.)
𝑧𝜑       ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 
Theoremsbco2d 2515 A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑 → Ⅎ𝑧𝜓)       (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
 
Theoremsbco3 2516 A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.) (New usage is discouraged.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
 
Theoremsbcom 2517 A commutativity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Check out sbcom3vv 2102 for a version requiring fewer axioms. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 20-Sep-2018.) (New usage is discouraged.)
([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)
 
Theoremsbtrt 2518 Partially closed form of sbtr 2519. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by BJ, 4-Jun-2019.) (New usage is discouraged.)
𝑦𝜑       (∀𝑦[𝑦 / 𝑥]𝜑𝜑)
 
Theoremsbtr 2519 A partial converse to sbt 2072. If the substitution of a variable for a nonfree one in a wff gives a theorem, then the original wff is a theorem. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by BJ, 15-Sep-2018.) (New usage is discouraged.)
𝑦𝜑    &   [𝑦 / 𝑥]𝜑       𝜑
 
Theoremsb8 2520 Substitution of variable in universal quantifier. Usage of this theorem is discouraged because it depends on ax-13 2371. For a version requiring disjoint variables, but fewer axioms, see sb8v 2353. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) (New usage is discouraged.)
𝑦𝜑       (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb8e 2521 Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2371. For a version requiring disjoint variables, but fewer axioms, see sb8ev 2354. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) (New usage is discouraged.)
𝑦𝜑       (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb9 2522 Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2523. (Revised by Wolf Lammen, 15-Jun-2019.) (New usage is discouraged.)
(∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb9i 2523 Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Jun-2019.) (New usage is discouraged.)
(∀𝑥[𝑥 / 𝑦]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsbhb 2524* Two ways of expressing "𝑥 is (effectively) not free in 𝜑". Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 29-May-2009.) (New usage is discouraged.)
((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
 
Theoremnfsbd 2525* Deduction version of nfsb 2526. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 15-Feb-2013.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑧𝜓)       (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
 
Theoremnfsb 2526* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Usage of this theorem is discouraged because it depends on ax-13 2371. For a version requiring more disjoint variables, but fewer axioms, see nfsbv 2329. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Feb-2024.) (New usage is discouraged.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑
 
TheoremnfsbOLD 2527* Obsolete version of nfsb 2526 as of 25-Feb-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑
 
Theoremhbsb 2528* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker hbsbw 2173 when possible. (Contributed by NM, 12-Aug-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
 
Theoremsb7f 2529* This version of dfsb7 2280 does not require that 𝜑 and 𝑧 be disjoint. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1918, i.e., that does not have the concept of a variable not occurring in a formula. (Definition dfsb1 2484 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
𝑧𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
 
Theoremsb7h 2530* This version of dfsb7 2280 does not require that 𝜑 and 𝑧 be disjoint. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1918, i.e., that does not have the concept of a variable not occurring in a formula. (Definition dfsb1 2484 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
 
Theoremsb10f 2531* Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
𝑥𝜑       ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑))
 
Theoremsbal1 2532* Check out sbal 2163 for a version not dependent on ax-13 2371. A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑧 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremsbal2 2533* Move quantifier in and out of substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Check out sbal 2163 for a version replacing the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 24-Dec-2022.) (Proof shortened by Wolf Lammen, 23-Sep-2023.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theorem2sb8e 2534* An equivalent expression for double existence. Usage of this theorem is discouraged because it depends on ax-13 2371. For a version requiring more disjoint variables, but fewer axioms, see 2sb8ev 2355. (Contributed by Wolf Lammen, 2-Nov-2019.) (New usage is discouraged.)
(∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
 
1.6  Uniqueness and unique existence
 
Theoremdfmoeu 2535* An elementary proof of moeu 2582 in disguise, connecting an expression characterizing uniqueness (df-mo 2539) to that of existential uniqueness (eu6 2573). No particular order of definition is required, as one can be derived from the other. This is shown here and in dfeumo 2536. (Contributed by Wolf Lammen, 27-May-2019.)
((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremdfeumo 2536* An elementary proof showing the reverse direction of dfmoeu 2535. Here the characterizing expression of existential uniqueness (eu6 2573) is derived from that of uniqueness (df-mo 2539). (Contributed by Wolf Lammen, 3-Oct-2023.)
((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
1.6.1  Uniqueness: the at-most-one quantifier
 
Syntaxwmo 2537 Extend wff definition to include the at-most-one quantifier ("there exists at most one 𝑥 such that 𝜑").
wff ∃*𝑥𝜑
 
Theoremmojust 2538* Soundness justification theorem for df-mo 2539 (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). (Contributed by NM, 11-Mar-2010.) Added this theorem by adapting the proof of eujust 2570. (Revised by BJ, 30-Sep-2022.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
 
Definitiondf-mo 2539* Define the at-most-one quantifier. The expression ∃*𝑥𝜑 is read "there exists at most one 𝑥 such that 𝜑". This is also called the "uniqueness quantifier" but that expression is also used for the unique existential quantifier df-eu 2568, therefore we avoid that ambiguous name.

Notation of [BellMachover] p. 460, whose definition we show as mo3 2563. For other possible definitions see moeu 2582 and mo4 2565. (Contributed by Wolf Lammen, 27-May-2019.) Make this the definition (which used to be moeu 2582, while this definition was then proved as dfmo 2595). (Revised by BJ, 30-Sep-2022.)

(∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremnexmo 2540 Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022.) Avoid ax-11 2158. (Revised by Wolf Lammen, 16-Oct-2022.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
 
Theoremexmo 2541 Any proposition holds for some 𝑥 or holds for at most one 𝑥. (Contributed by NM, 8-Mar-1995.) Shorten proof and avoid df-eu 2568. (Revised by BJ, 14-Oct-2022.)
(∃𝑥𝜑 ∨ ∃*𝑥𝜑)
 
Theoremmoabs 2542 Absorption of existence condition by uniqueness. (Contributed by NM, 4-Nov-2002.) Shorten proof and avoid df-eu 2568. (Revised by BJ, 14-Oct-2022.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
 
Theoremmoim 2543 The at-most-one quantifier reverses implication. (Contributed by NM, 22-Apr-1995.)
(∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
 
Theoremmoimi 2544 The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006.) Remove use of ax-5 1918. (Revised by Steven Nguyen, 9-May-2023.)
(𝜑𝜓)       (∃*𝑥𝜓 → ∃*𝑥𝜑)
 
Theoremmoimdv 2545* The at-most-one quantifier reverses implication, deduction form. (Contributed by Thierry Arnoux, 25-Feb-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜒 → ∃*𝑥𝜓))
 
Theoremmobi 2546 Equivalence theorem for the at-most-one quantifier. (Contributed by BJ, 7-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)
(∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
 
Theoremmobii 2547 Formula-building rule for the at-most-one quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) Avoid ax-5 1918. (Revised by Wolf Lammen, 24-Sep-2023.)
(𝜓𝜒)       (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)
 
Theoremmobidv 2548* Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
 
Theoremmobid 2549 Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2141, ax-11 2158, ax-13 2371. (Revised by BJ, 14-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
 
Theoremmoa1 2550 If an implication holds for at most one value, then its consequent holds for at most one value. See also ala1 1821 and exa1 1845. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (Revised by BJ, 29-Mar-2021.)
(∃*𝑥(𝜑𝜓) → ∃*𝑥𝜓)
 
Theoremmoan 2551 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
(∃*𝑥𝜑 → ∃*𝑥(𝜓𝜑))
 
Theoremmoani 2552 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
∃*𝑥𝜑       ∃*𝑥(𝜓𝜑)
 
Theoremmoor 2553 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
(∃*𝑥(𝜑𝜓) → ∃*𝑥𝜑)
 
Theoremmooran1 2554 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑𝜓))
 
Theoremmooran2 2555 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃*𝑥(𝜑𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))
 
Theoremnfmo1 2556 Bound-variable hypothesis builder for the at-most-one quantifier. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) Adapt to new definition. (Revised by BJ, 1-Oct-2022.)
𝑥∃*𝑥𝜑
 
Theoremnfmod2 2557 Bound-variable hypothesis builder for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2371. See nfmodv 2558 for a version replacing the distinctor with a disjoint variable condition, not requiring ax-13 2371. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid df-eu 2568. (Revised by BJ, 14-Oct-2022.) (New usage is discouraged.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
 
Theoremnfmodv 2558* Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2560 for a version without disjoint variable conditions but requiring ax-13 2371. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by BJ, 28-Jan-2023.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
 
Theoremnfmov 2559* Bound-variable hypothesis builder for the at-most-one quantifier. See nfmo 2561 for a version without disjoint variable conditions but requiring ax-13 2371. (Contributed by NM, 9-Mar-1995.) (Revised by Wolf Lammen, 2-Oct-2023.)
𝑥𝜑       𝑥∃*𝑦𝜑
 
Theoremnfmod 2560 Bound-variable hypothesis builder for the at-most-one quantifier. Deduction version of nfmo 2561. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfmodv 2558 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
 
Theoremnfmo 2561 Bound-variable hypothesis builder for the at-most-one quantifier. Note that 𝑥 and 𝑦 need not be disjoint. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfmov 2559 when possible. (Contributed by NM, 9-Mar-1995.) (New usage is discouraged.)
𝑥𝜑       𝑥∃*𝑦𝜑
 
Theoremmof 2562* Version of df-mo 2539 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2595 from this proof, and prove mof 2562 from it (as of 30-Sep-2022, directly from df-mo 2539). (Revised by Wolf Lammen, 28-May-2019.) Avoid ax-13 2371. (Revised by Wolf Lammen, 16-Oct-2022.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremmo3 2563* Alternate definition of the at-most-one quantifier. Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) Remove dependency on ax-13 2371. (Revised by BJ and WL, 29-Jan-2023.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
 
Theoremmo 2564* Equivalent definitions of "there exists at most one". (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)
𝑦𝜑       (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
 
Theoremmo4 2565* At-most-one quantifier expressed using implicit substitution. This theorem is also a direct consequence of mo4f 2566, but this proof is based on fewer axioms.

By the way, swapping 𝑥, 𝑦 and 𝜑, 𝜓 leads to an expression for ∃*𝑦𝜓, which is equivalent to ∃*𝑥𝜑 (is a proof line), so the right hand side is a rare instance of an expression where swapping the quantifiers can be done without ax-11 2158. (Contributed by NM, 26-Jul-1995.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2023.)

(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
 
Theoremmo4f 2566* At-most-one quantifier expressed using implicit substitution. Note that the disjoint variable condition on 𝑦, 𝜑 can be replaced by the nonfreeness hypothesis 𝑦𝜑 with essentially the same proof. (Contributed by NM, 10-Apr-2004.) Remove dependency on ax-13 2371. (Revised by Wolf Lammen, 19-Jan-2023.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
 
1.6.2  Unique existence: the unique existential quantifier
 
Syntaxweu 2567 Extend wff definition to include the unique existential quantifier ("there exists a unique 𝑥 such that 𝜑").
wff ∃!𝑥𝜑
 
Definitiondf-eu 2568 Define the existential uniqueness quantifier. This expresses unique existence, or existential uniqueness, which is the conjunction of existence (df-ex 1788) and uniqueness (df-mo 2539). The expression ∃!𝑥𝜑 is read "there exists exactly one 𝑥 such that 𝜑 " or "there exists a unique 𝑥 such that 𝜑". This is also called the "uniqueness quantifier" but that expression is also used for the at-most-one quantifier df-mo 2539, therefore we avoid that ambiguous name.

Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2611, eu2 2610, eu3v 2569, and eu6 2573. As for double unique existence, beware that the expression ∃!𝑥∃!𝑦𝜑 means "there exists a unique 𝑥 such that there exists a unique 𝑦 such that 𝜑 " which is a weaker property than "there exists exactly one 𝑥 and one 𝑦 such that 𝜑 " (see 2eu4 2655). (Contributed by NM, 12-Aug-1993.) Make this the definition (which used to be eu6 2573, while this definition was then proved as dfeu 2594). (Revised by BJ, 30-Sep-2022.)

(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
 
Theoremeu3v 2569* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Replace a nonfreeness hypothesis with a disjoint variable condition on 𝜑, 𝑦 to reduce axiom usage. (Revised by Wolf Lammen, 29-May-2019.)
(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theoremeujust 2570* Soundness justification theorem for eu6 2573 when this was the definition of the unique existential quantifier (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). See eujustALT 2571 for a proof that provides an example of how it can be achieved through the use of dvelim 2450. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
 
TheoremeujustALT 2571* Alternate proof of eujust 2570 illustrating the use of dvelim 2450. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
 
Theoremeu6lem 2572* Lemma of eu6im 2574. A dissection of an idiom characterizing existential uniqueness. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2568 was then proved as dfeu 2594. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Extract common proof lines. (Revised by Wolf Lammen, 3-Mar-2023.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
 
Theoremeu6 2573* Alternate definition of the unique existential quantifier df-eu 2568 not using the at-most-one quantifier. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2568 was then proved as dfeu 2594. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Remove use of ax-11 2158. (Revised by SN, 21-Sep-2023.)
(∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremeu6im 2574* One direction of eu6 2573 needs fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃!𝑥𝜑)
 
Theoremeuf 2575* Version of eu6 2573 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.) Avoid ax-13 2371. (Revised by Wolf Lammen, 16-Oct-2022.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremeuex 2576 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.)
(∃!𝑥𝜑 → ∃𝑥𝜑)
 
Theoremeumo 2577 Existential uniqueness implies uniqueness. (Contributed by NM, 23-Mar-1995.)
(∃!𝑥𝜑 → ∃*𝑥𝜑)
 
Theoremeumoi 2578 Uniqueness inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
∃!𝑥𝜑       ∃*𝑥𝜑
 
Theoremexmoeub 2579 Existence implies that uniqueness is equivalent to unique existence. (Contributed by NM, 5-Apr-2004.)
(∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
 
Theoremexmoeu 2580 Existence is equivalent to uniqueness implying existential uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.)
(∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))
 
Theoremmoeuex 2581 Uniqueness implies that existence is equivalent to unique existence. (Contributed by BJ, 7-Oct-2022.)
(∃*𝑥𝜑 → (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
 
Theoremmoeu 2582 Uniqueness is equivalent to existence implying unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by NM, 8-Mar-1995.) This used to be the definition of the at-most-one quantifier, while df-mo 2539 was then proved as dfmo 2595. (Revised by BJ, 30-Sep-2022.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
 
Theoremeubi 2583 Equivalence theorem for the unique existential quantifier. Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) Reduce dependencies on axioms. (Revised by BJ, 7-Oct-2022.)
(∀𝑥(𝜑𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))
 
Theoremeubii 2584 Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) Avoid ax-5 1918. (Revised by Wolf Lammen, 27-Sep-2023.)
(𝜑𝜓)       (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)
 
Theoremeubidv 2585* Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 
Theoremeubid 2586 Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 19-Feb-2023.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 
Theoremnfeu1 2587 Bound-variable hypothesis builder for uniqueness. See also nfeu1ALT 2588. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃!𝑥𝜑
 
Theoremnfeu1ALT 2588 Alternate proof of nfeu1 2587. This illustrates the systematic way of proving nonfreeness in a defined expression: consider the definiens as a tree whose nodes are its subformulas, and prove by tree-induction nonfreeness of each node, starting from the leaves (generally using nfv 1922 or nf* theorems for previously defined expressions) and up to the root. Here, the definiens is a conjunction of two previously defined expressions, which automatically yields the present proof. (Contributed by BJ, 2-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥∃!𝑥𝜑
 
Theoremnfeud2 2589 Bound-variable hypothesis builder for uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2371. Check out nfeudw 2590 for a version that replaces the distinctor with a disjoint variable condition, not requiring ax-13 2371. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof shortened by BJ, 14-Oct-2022.) (New usage is discouraged.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 
Theoremnfeudw 2590* Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2593. Version of nfeud 2591 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 
Theoremnfeud 2591 Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2593. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfeudw 2590 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 
Theoremnfeuw 2592* Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2593 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 8-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑥𝜑       𝑥∃!𝑦𝜑
 
Theoremnfeu 2593 Bound-variable hypothesis builder for the unique existential quantifier. Note that 𝑥 and 𝑦 need not be disjoint. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfeuw 2592 when possible. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)
𝑥𝜑       𝑥∃!𝑦𝜑
 
Theoremdfeu 2594 Rederive df-eu 2568 from the old definition eu6 2573. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.) (Proof shortened by BJ, 7-Oct-2022.) (Proof modification is discouraged.) Use df-eu 2568 instead. (New usage is discouraged.)
(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
 
Theoremdfmo 2595* Rederive df-mo 2539 from the old definition moeu 2582. (Contributed by Wolf Lammen, 27-May-2019.) (Proof modification is discouraged.) Use df-mo 2539 instead. (New usage is discouraged.)
(∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremeuequ 2596* There exists a unique set equal to a given set. Special case of eueqi 3622 proved using only predicate calculus. The proof needs 𝑦 = 𝑧 be free of 𝑥. This is ensured by having 𝑥 and 𝑦 be distinct. Alternately, a distinctor ¬ ∀𝑥𝑥 = 𝑦 could have been used instead. See eueq 3621 and eueqi 3622 for classes. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.) Reduce axiom usage. (Revised by Wolf Lammen, 1-Mar-2023.)
∃!𝑥 𝑥 = 𝑦
 
Theoremsb8eulem 2597* Lemma. Factor out the common proof skeleton of sb8euv 2598 and sb8eu 2599. Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) Factor out common proof lines. (Revised by Wolf Lammen, 9-Feb-2023.)
𝑦[𝑤 / 𝑥]𝜑       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb8euv 2598* Variable substitution in unique existential quantifier. Version of sb8eu 2599 requiring more disjoint variables, but fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Wolf Lammen, 7-Feb-2023.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb8eu 2599 Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2371. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2598. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) (New usage is discouraged.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb8mo 2600 Variable substitution for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46180
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