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Theorem List for Metamath Proof Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem19.21tOLD 2201 Obsolete proof of 19.21t 2061 as of 6-Oct-2021. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theorem19.21OLD 2202 Obsolete proof of 19.21 2062 as of 6-Oct-2021. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theorem19.21-2OLD 2203 Obsolete proof of 19.21-2 2065 as of 6-Oct-2021. (Contributed by NM, 4-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))

Theorem19.21hOLD 2204 Obsolete proof of 19.21h 2107 as of 6-Oct-2021. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theoremstdpc5OLDOLD 2205 Obsolete proof of stdpc5 2063 as of 6-Oct-2021. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 4-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theorem19.23tOLD 2206 Obsolete proof of 19.23t 2066 as of 6-Oct-2021. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Theorem19.23OLD 2207 Obsolete proof of 19.23 2067 as of 6-Oct-2021. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.23hOLD 2208 Obsolete proof of 19.23h 2108 as of 6-Oct-2021. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

TheoremexlimiOLD 2209 Obsolete proof of exlimi 2073 as of 6-Oct-2021. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

TheoremexlimihOLD 2210 Obsolete proof of exlimih 2133 as of 6-Oct-2021. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

TheoremexlimdOLD 2211 Obsolete proof of exlimd 2074 as of 6-Oct-2021. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

TheoremexlimdhOLD 2212 Obsolete proof of exlimdh 2134 as of 6-Oct-2021. (Contributed by NM, 28-Jan-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

TheoremnfdiOLD 2213 Obsolete proof of nf5di 2105 as of 6-Oct-2021. (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜑)       𝑥𝜑

TheoremnfimdOLD 2214 Obsolete proof of nfimd 1812 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremhbim1OLD 2215 Obsolete proof of hbim 2112 as of 6-Oct-2021. (Contributed by NM, 2-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremnfim1OLD 2216 Obsolete proof of nfim1 2055 as of 6-Oct-2021. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

TheoremnfimOLD 2217 Obsolete proof of nfim 1813 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

TheoremhbimdOLD 2218 Obsolete proof of hbimd 2111 as of 6-Oct-2021. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))

TheoremhbimOLD 2219 Obsolete proof of hbim 2112 as of 6-Oct-2021. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

TheoremnfandOLD 2220 Obsolete proof of nfand 1814 as of 6-Oct-2021. (Contributed by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremnf3andOLD 2221 Obsolete proof of nf3and 1815 as of 6-Oct-2021. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑥𝜃)       (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))

Theorem19.27OLD 2222 Obsolete proof of 19.27 2082 as of 6-Oct-2021. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.28OLD 2223 Obsolete proof of 19.28 2083 as of 6-Oct-2021. (Contributed by NM, 1-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theoremnfan1OLD 2224 Obsolete proof of nfan1 2056 as of 6-Oct-2021. (Contributed by Mario Carneiro, 3-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

TheoremnfanOLDOLD 2225 Obsolete proof of nfan 1816 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

TheoremnfnanOLD 2226 Obsolete proof of nfnan 1818 as of 6-Oct-2021. (Contributed by Scott Fenton, 2-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremnf3anOLD 2227 Obsolete proof of nf3an 1819 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)

TheoremhbanOLD 2228 Obsolete proof of hban 2113 as of 6-Oct-2021. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhb3anOLD 2229 Obsolete proof of hb3an 2114 as of 6-Oct-2021. (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))

TheoremnfbidOLD 2230 Obsolete proof of nfbid 1820 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

TheoremnfbiOLD 2231 Obsolete proof of nfbi 1821 as of 6-Oct-2021. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

TheoremnforOLD 2232 Obsolete proof of nfor 1822 as of 6-Oct-2021. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremnf3orOLD 2233 Obsolete proof of nf3or 1823 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)

1.5.4  Axiom scheme ax-13 (Quantified Equality)

Axiomax-13 2234 Axiom of Quantified Equality. One of the equality and substitution axioms of predicate calculus with equality.

An equivalent way to express this axiom that may be easier to understand is 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧))) (see ax13b 1951). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent ¬ 𝑥 = 𝑦 to hold, 𝑥 and 𝑦 must have different values and thus cannot be the same object-language variable (so they are effectively "distinct variables" even though no \$d is present). Similarly, 𝑥 and 𝑧 cannot be the same object-language variable. Therefore, 𝑥 will not occur in the wff 𝑦 = 𝑧 when the first two antecedents hold, so analogous to ax-5 1827, the conclusion (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧) follows. Note that ax-5 1827 cannot prove this because its distinct variable (\$d) requirement is not satisfied directly but only indirectly (outside of Metamath) by the argument above.

The original version of this axiom was ax-c9 33193 and was replaced with this shorter ax-13 2234 in December 2015. The old axiom is proved from this one as theorem axc9 2290.

The primary purpose of this axiom is to provide a way to introduce the quantifier 𝑥 on 𝑦 = 𝑧 even when 𝑥 and 𝑦 are substituted with the same variable. In this case, the first antecedent becomes ¬ 𝑥 = 𝑥 and the axiom still holds.

Although this version is shorter, the original version axc9 2290 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of axc9 2290 is in dvelimh 2324 which converts a distinct variable pair to the distinctor antecedent ¬ ∀𝑥𝑥 = 𝑦. In particular, it is conjectured that it is not possible to prove ax6 2239 from ax6v 1876 without this axiom.

This axiom can be weakened if desired by adding distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. To show that, we add these restrictions to theorem ax13v 2235 and use only ax13v 2235 for further derivations. Thus, ax13v 2235 should be the only theorem referencing this axiom. Other theorems can reference either ax13v 2235 (preferred) or ax13 2237 (if the stronger form is needed).

This axiom scheme is logically redundant (see ax13w 2000) but is used as an auxiliary axiom scheme to achieve scheme completeness (i.e. so that all possible cases of bundling can be proved; see text linked at mmtheorems.html#ax6dgen). It is not known whether this axiom can be derived from the others. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax13v 2235* A weaker version of ax-13 2234 with distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. In order to show (with ax13 2237) that this weakening is still adequate, this should be the only theorem referencing ax-13 2234 directly.

Had we additonally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1827. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. (Contributed by NM, 30-Jun-2016.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax13lem1 2236* A version of ax13v 2235 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2237 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremax13 2237 Derive ax-13 2234 from ax13v 2235 and Tarski's FOL. This shows that the weakening in ax13v 2235 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) Reduce axiom usage (Revised by Wolf Lammen, 2-Jun-2021.)
𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax6e 2238 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-4 1728 through ax-9 1986, all axioms other than ax-6 1875 are believed to be theorems of free logic, although the system without ax-6 1875 is not complete in free logic.

It is preferred to use ax6ev 1877 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2236 became available. (Revised by Wolf Lammen, 8-Sep-2018.)

𝑥 𝑥 = 𝑦

Theoremax6 2239 Theorem showing that ax-6 1875 follows from the weaker version ax6v 1876. (Even though this theorem depends on ax-6 1875, all references of ax-6 1875 are made via ax6v 1876. An earlier version stated ax6v 1876 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1875 so that all proofs can be traced back to ax6v 1876. When possible, use the weaker ax6v 1876 rather than ax6 2239 since the ax6v 1876 derivation is much shorter and requires fewer axioms. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 4-Feb-2018.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦

Theoremaxc10 2240 Show that the original axiom ax-c10 33189 can be derived from ax6 2239 and others. See ax6fromc10 33199 for the rederivation of ax6 2239 from ax-c10 33189.

Normally, axc10 2240 should be used rather than ax-c10 33189, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Theoremspimt 2241 Closed theorem form of spim 2242. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))

Theoremspim 2242 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2242 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimed 2243 Deduction version of spime 2244. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)
(𝜒 → Ⅎ𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (𝜑 → ∃𝑥𝜓))

Theoremspime 2244 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspimv 2245* A version of spim 2242 with a distinct variable requirement instead of a bound variable hypothesis. See also spimv1 2101 and spimvw 1914. See also spimvALT 2246. (Contributed by NM, 31-Jul-1993.) Removed dependency on ax-10 2006. (Revised by BJ, 29-Nov-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

TheoremspimvALT 2246* Alternate proof of spimv 2245. Shorter but requires more axioms. (Contributed by NM, 31-Jul-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimev 2247* Distinct-variable version of spime 2244. (Contributed by NM, 10-Jan-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspv 2248* Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspei 2249 Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑

Theoremchvar 2250 Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theoremchvarv 2251* Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by NM, 20-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 14-Jul-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

TheoremchvarvOLD 2252* Obsolete proof of chvarv 2251 as of 14-Jul-2021. (Contributed by NM, 20-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theoremcbv3 2253 Rule used to change bound variables, using implicit substitution, that does not use ax-c9 33193. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbv3h 2254 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 8-Jun-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)
(𝜑 → ∀𝑦𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbv1 2255 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theoremcbv1h 2256 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)
(𝜑 → (𝜓 → ∀𝑦𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theoremcbv2h 2257 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓 → ∀𝑦𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbv2 2258 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbval 2259 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvex 2260 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 21-Jun-1993.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbvalv 2261* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 17-Jul-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

TheoremcbvalvOLD 2262* Obsolete proof of cbvalv 2261 as of 17-Jul-2021. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvexv 2263* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 17-Jul-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

TheoremcbvexvOLD 2264* Obsolete proof of cbvexv 2263 as of 17-Jul-2021. (Contributed by NM, 21-Jun-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbvald 2265* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2325. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvexd 2266* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2325. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbval2 2267* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2 2268* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 16-Jun-2019.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbvaldva 2269* Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 18-Jul-2021.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

TheoremcbvaldvaOLD 2270* Obsolete proof of cbvaldva 2269 as of 18-Jul-2021. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvexdva 2271* Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 18-Jul-2021.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

TheoremcbvexdvaOLD 2272* Obsolete proof of cbvexdva 2271 as of 18-Jul-2021. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbval2v 2273* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 18-Jul-2021.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbval2vOLD 2274* Obsolete proof of cbval2v 2273 as of 18-Jul-2021. (Contributed by NM, 4-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2v 2275* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 18-Jul-2021.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbvex2vOLD 2276* Obsolete proof of cbvex2v 2275 as of 18-Jul-2021. (Contributed by NM, 26-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbvex4v 2277* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))    &   ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))       (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)

Theoremequs4 2278 Lemma used in proofs of implicit substitution properties. The converse requires either a dv condition (sb56 2136) or a non-freeness hypothesis (equs45f 2338). See equs4v 1917 for a version requiring fewer axioms. (Contributed by NM, 10-May-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremequsal 2279 An equivalence related to implicit substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsalh 2280 An equivalence related to implicit substitution. (Contributed by NM, 2-Jun-1993.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsex 2281 An equivalence related to implicit substitution. See equsexv 2095 for a version with a dv condition which does not require ax-13 2234. See equsexALT 2282 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

TheoremequsexALT 2282 Alternate proof of equsex 2281. This proves the result directly, instead of as a corollary of equsal 2279 via equs4 2278. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2234 is ax6e 2238. This proof mimics that of equsal 2279 (in particular, note pm5.32i 667, exbii 1764, 19.41 2090, mpbiran 955 correspond respectively to pm5.74i 259, albii 1737, 19.23 2067, a1bi 351). (Contributed by BJ, 20-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexh 2283 An equivalence related to implicit substitution. See equsexhv 2135 for a version with a dv condition which does not require ax-13 2234. (Contributed by NM, 5-Aug-1993.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremax13lem2 2284* Lemma for nfeqf2 2285. This lemma is equivalent to ax13v 2235 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 8-Sep-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2020.) (New usage is discouraged.)
𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))

Theoremnfeqf2 2285* An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 9-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)

Theoremdveeq2 2286* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 2021. (Revised by Wolf Lammen, 8-Sep-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremnfeqf1 2287* An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 10-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)

Theoremdveeq1 2288* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) Remove dependency on ax-11 2021. (Revised by Wolf Lammen, 8-Sep-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremnfeqf 2289 A variable is effectively not free in an equality if it is not either of the involved variables. version of ax-c9 33193. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2021. (Revised by Wolf Lammen, 6-Sep-2018.)
((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)

Theoremaxc9 2290 Derive set.mm's original ax-c9 33193 from the shorter ax-13 2234. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.)
(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Theoremaxc15 2291 Derivation of set.mm's original ax-c15 33192 from ax-c11n 33191 and the shorter ax-12 2034 that has replaced it.

Theorem ax12 2292 shows the reverse derivation of ax-12 2034 from ax-c15 33192.

Normally, axc15 2291 should be used rather than ax-c15 33192, except by theorems specifically studying the latter's properties. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12 2292 Rederivation of axiom ax-12 2034 from ax12v 2035, axc11r 2175, and other axioms. (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof shortened by BJ, 4-Jul-2021.)
(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax13OLD 2293 Obsolete proof of ax13 2237 as of 2-Jun-2021. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremaxc11nlemALT 2294* Alternate version of axc11nlemOLD2 1975 used in an older proof. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑤 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11n 2295 Derive set.mm's original ax-c11n 33191 from others. Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). If a disjoint variable condition is added on 𝑥 and 𝑦, then this becomes an instance of aevlem 1968. Use aecom 2299 instead when this does not lengthen the proof. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Revised by Wolf Lammen, 30-Nov-2019.) (Proof shortened by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 2-Jul-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11nOLD 2296 Obsolete proof of axc11n 2295 as of 2-Jul-2021. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Revised by Wolf Lammen, 30-Nov-2019.) (Proof shortened by BJ, 29-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11nOLDOLD 2297 Old proof of axc11n 2295. Obsolete as of 29-Mar-2021. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Adapt to a modification of axc11nlemOLD2 1975. (Revised by Wolf Lammen, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11nALT 2298 Alternate proof of axc11n 2295 from axc11nlemALT 2294. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaecom 2299 Commutation law for identical variable specifiers. Both sides of the biconditional are true when 𝑥 and 𝑦 are substituted with the same variable. (Contributed by NM, 10-May-1993.) Changed to a biconditional. (Revised by BJ, 26-Sep-2019.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)

Theoremaecoms 2300 A commutation rule for identical variable specifiers. (Contributed by NM, 10-May-1993.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

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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 42301-42360
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