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Theorem List for Metamath Proof Explorer - 34001-34100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-cbveximi 34001* An equality-free general instance of one half of a precise form of bj-cbvex 34003. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
(𝜒 → (𝜑𝜓))    &   𝑥𝑦𝜒       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-cbval 34002* Changing a bound variable (universal quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
𝑦𝑥 𝑥 = 𝑦    &   𝑥𝑦 𝑦 = 𝑥    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑦 = 𝑥𝑥 = 𝑦)       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theorembj-cbvex 34003* Changing a bound variable (existential quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
𝑦𝑥 𝑥 = 𝑦    &   𝑥𝑦 𝑦 = 𝑥    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑦 = 𝑥𝑥 = 𝑦)       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntaxwmoo 34004 Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑

Definitiondf-bj-mo 34005* Definition of the uniqueness quantifier which is correct on the empty domain. Instead of the fresh variable 𝑧, one could save a dummy variable by using 𝑥 or 𝑦 at the cost of having nested quantifiers on the same variable. (Contributed by BJ, 12-Mar-2023.)
(∃**𝑥𝜑 ↔ ∀𝑧𝑦𝑥(𝜑𝑥 = 𝑦))

20.15.4.4  Equality and substitution

Theorembj-ssbeq 34006* Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 1971. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 34006 first with a DV condition on 𝑥, 𝑡, and then in the general case. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
([𝑡 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧)

Theorembj-ssblem1 34007* A lemma for the definiens of df-sb 2071. An instance of sp 2184 proved without it. Note: it has a common subproof with sbjust 2069. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssblem2 34008* An instance of ax-11 2162 proved without it. The converse may not be provable without ax-11 2162 (since using alcomiw 2051 would require a DV on 𝜑, 𝑥, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))

Theorembj-ax12v 34009* A weaker form of ax-12 2179 and ax12v 2180, namely the generalization over 𝑥 of the latter. In this statement, all occurrences of 𝑥 are bound. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))

Theorembj-ax12 34010* Remove a DV condition from bj-ax12v 34009 (using core axioms only). (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))

Theorembj-ax12ssb 34011* The axiom bj-ax12 34010 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
[𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑)

Theorembj-19.41al 34012 Special case of 19.41 2239 proved from Tarski, ax-10 2146 (modal5) and hba1 2303 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))

Theorembj-equsexval 34013* Special case of equsexv 2271 proved from Tarski, ax-10 2146 (modal5) and hba1 2303 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜓)

Theorembj-sb56 34014* Proof of sb56 2279 from Tarski, ax-10 2146 (modal5) and bj-ax12 34010. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-ssbid2 34015 A special case of sbequ2 2252. (Contributed by BJ, 22-Dec-2020.)
([𝑥 / 𝑥]𝜑𝜑)

Theorembj-ssbid2ALT 34016 Alternate proof of bj-ssbid2 34015, not using sbequ2 2252. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑥 / 𝑥]𝜑𝜑)

Theorembj-ssbid1 34017 A special case of sbequ1 2251. (Contributed by BJ, 22-Dec-2020.)
(𝜑 → [𝑥 / 𝑥]𝜑)

Theorembj-ssbid1ALT 34018 Alternate proof of bj-ssbid1 34017, not using sbequ1 2251. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → [𝑥 / 𝑥]𝜑)

Theorembj-ax6elem1 34019* Lemma for bj-ax6e 34021. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theorembj-ax6elem2 34020* Lemma for bj-ax6e 34021. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
(∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)

Theorembj-ax6e 34021 Proof of ax6e 2403 (hence ax6 2404) from Tarski's system, ax-c9 36091, ax-c16 36093. Remark: ax-6 1971 is used only via its principal (unbundled) instance ax6v 1972. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦

Theorembj-spimvwt 34022* Closed form of spimvw 2003. See also spimt 2406. (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑𝜓))

Theorembj-spnfw 34023 Theorem close to a closed form of spnfw 1985. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorembj-cbvexiw 34024* Change bound variable. This is to cbvexvw 2045 what cbvaliw 2014 is to cbvalvw 2044. TODO: move after cbvalivw 2015. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-cbvexivw 34025* Change bound variable. This is to cbvexvw 2045 what cbvalivw 2015 is to cbvalvw 2044. TODO: move after cbvalivw 2015. (Contributed by BJ, 17-Mar-2020.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)

Theorembj-modald 34026 A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Theorembj-denot 34027* A weakening of ax-6 1971 and ax6v 1972. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
(𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)

Theorembj-eqs 34028* A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2392. The disjoint variable condition DV (𝑥, 𝜑) is necessary for both directions: consider substituting 𝑥 = 𝑧 for 𝜑. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-cbvexw 34029* Change bound variable. This is to cbvexvw 2045 what cbvalw 2043 is to cbvalvw 2044. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (∃𝑦𝑥𝜑 → ∃𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theorembj-ax12w 34030* The general statement that ax12w 2138 proves. (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))    &   (𝑦 = 𝑧 → (𝜓𝜃))       (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑𝜓)))

20.15.4.7  Membership predicate, ax-8 and ax-9

Theorembj-ax89 34031 A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2117 and ax-9 2125. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2117 and ax-9 2125, as proved here. In the other direction, one can prove ax-8 2117 (respectively ax-9 2125) from bj-ax89 34031 by using mpan2 690 (respectively mpan 689) and equid 2020. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Theorembj-elequ12 34032 An identity law for the non-logical predicate, which combines elequ1 2122 and elequ2 2130. For the analogous theorems for class terms, see eleq1 2903, eleq2 2904 and eleq12 2905. TODO: move to main part. (Contributed by BJ, 29-Sep-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Theorembj-cleljusti 34033* One direction of cleljust 2124, requiring only ax-1 6-- ax-5 1912 and ax8v1 2119. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
(∃𝑧(𝑧 = 𝑥𝑧𝑦) → 𝑥𝑦)

Theorembj-alcomexcom 34034 Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1811 section, soon after 2nexaln 1831, and used to prove excom 2170. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))

Theorembj-hbalt 34035 Closed form of hbal 2175. When in main part, prove hbal 2175 and hbald 2176 from it. (Contributed by BJ, 2-May-2019.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))

Theoremaxc11n11 34036 Proof of axc11n 2450 from { ax-1 6-- ax-7 2016, axc11 2454 } . Almost identical to axc11nfromc11 36127. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11n11r 34037 Proof of axc11n 2450 from { ax-1 6-- ax-7 2016, axc9 2402, axc11r 2388 } (note that axc16 2264 is provable from { ax-1 6-- ax-7 2016, axc11r 2388 }).

Note that axc11n 2450 proves (over minimal calculus) that axc11 2454 and axc11r 2388 are equivalent. Therefore, axc11n11 34036 and axc11n11r 34037 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2454 appears slightly stronger since axc11n11r 34037 requires axc9 2402 while axc11n11 34036 does not).

(Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theorembj-axc16g16 34038* Proof of axc16g 2263 from { ax-1 6-- ax-7 2016, axc16 2264 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theorembj-ax12v3 34039* A weak version of ax-12 2179 which is stronger than ax12v 2180. Note that if one assumes reflexivity of equality 𝑥 = 𝑥 (equid 2020), then bj-ax12v3 34039 implies ax-5 1912 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 34040. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ax12v3ALT 34040* Alternate proof of bj-ax12v3 34039. Uses axc11r 2388 and axc15 2446 instead of ax-12 2179. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-sb 34041* A weak variant of sbid2 2552 not requiring ax-13 2392 nor ax-10 2146. On top of Tarski's FOL, one implication requires only ax12v 2180, and the other requires only sp 2184. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-modalbe 34042 The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2340. (Contributed by BJ, 20-Oct-2019.)
(𝜑 → ∀𝑥𝑥𝜑)

Theorembj-spst 34043 Closed form of sps 2186. Once in main part, prove sps 2186 and spsd 2188 from it. (Contributed by BJ, 20-Oct-2019.)
((𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorembj-19.21bit 34044 Closed form of 19.21bi 2190. (Contributed by BJ, 20-Oct-2019.)
((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))

Theorembj-19.23bit 34045 Closed form of 19.23bi 2192. (Contributed by BJ, 20-Oct-2019.)
((∃𝑥𝜑𝜓) → (𝜑𝜓))

Theorembj-nexrt 34046 Closed form of nexr 2193. Contrapositive of 19.8a 2182. (Contributed by BJ, 20-Oct-2019.)
(¬ ∃𝑥𝜑 → ¬ 𝜑)

Theorembj-alrim 34047 Closed form of alrimi 2215. (Contributed by BJ, 2-May-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Theorembj-alrim2 34048 Uncurried (imported) form of bj-alrim 34047. (Contributed by BJ, 2-May-2019.)
((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (𝜑 → ∀𝑥𝜓))

Theorembj-nfdt0 34049 A theorem close to a closed form of nf5d 2294 and nf5dh 2152. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))

Theorembj-nfdt 34050 Closed form of nf5d 2294 and nf5dh 2152. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))

Theorembj-nexdt 34051 Closed form of nexd 2225. (Contributed by BJ, 20-Oct-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))

Theorembj-nexdvt 34052* Closed form of nexdv 1938. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))

Theorembj-alexbiex 34053 Adding a second quantifier is a tranparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∃𝑥𝜑)

Theorembj-exexbiex 34054 Adding a second quantifier is a tranparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∃𝑥𝜑)

Theorembj-alalbial 34055 Adding a second quantifier is a tranparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∀𝑥𝜑)

Theorembj-exalbial 34056 Adding a second quantifier is a tranparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∀𝑥𝜑)

Theorembj-19.9htbi 34057 Strengthening 19.9ht 2341 by replacing its succedent with a biconditional (19.9t 2206 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Theorembj-hbntbi 34058 Strengthening hbnt 2304 by replacing its succedent with a biconditional. See also hbntg 33070 and hbntal 41110. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 34057. (Proof modification is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))

Theorembj-biexal1 34059 A general FOL biconditional that generalizes 19.9ht 2341 among others. For this and the following theorems, see also 19.35 1879, 19.21 2209, 19.23 2213. When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-biexal2 34060 When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(∃𝑥𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-biexal3 34061 When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑𝜓))

Theorembj-bialal 34062 When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(∀𝑥𝜑𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))

Theorembj-biexex 34063 When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓))

Theorembj-hbext 34064 Closed form of hbex 2346. (Contributed by BJ, 10-Oct-2019.)
(∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Theorembj-nfalt 34065 Closed form of nfal 2344. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Theorembj-nfext 34066 Closed form of nfex 2345. (Contributed by BJ, 10-Oct-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Theorembj-eeanvw 34067* Version of exdistrv 1957 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2162. (The same can be done with eeeanv 2373 and ee4anv 2374.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Theorembj-modal4 34068 First-order logic form of the modal axiom (4). See hba1 2303. This is the standard proof of the implication in modal logic (B5 4). Its dual statement is bj-modal4e 34069. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Theorembj-modal4e 34069 First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 34068 (hba1 2303). (Contributed by BJ, 21-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥𝑥𝜑 → ∃𝑥𝜑)

Theorembj-modalb 34070 A short form of the axiom B of modal logic using only primitive symbols (→ , ¬ , ∀). (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theorembj-wnf1 34071 When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-wnf2 34072 When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
(∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-wnfanf 34073 When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the universal form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))

Theorembj-wnfenf 34074 When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the existential form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑𝜓))

Theorembj-subst 34075 Equivalent form of the axiom of substitution bj-ax12 34010. Although both sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 34039 on 𝑡, 𝜑) to hold, their equivalence holds without DV conditions. The forward implication is proved in modal (K4) while the reverse implication is proved in modal (T5). The LHS has the advantage of not involving nested quantifiers on the same variable. Its metaweakening is proved from the core axiom schemes in bj-substw 34076. Note that in the LHS, the reverse implication holds by equs4 2440 (or equs4v 2007 if a DV condition is added on 𝑥, 𝑡 as in bj-ax12 34010).

The LHS can be read as saying that if there exists a setvar equal to a given term witnessing 𝜑, then all setvars equal to that term also witness 𝜑. An equivalent suggestive form for the LHS is ¬ (∃𝑥(𝑥 = 𝑡𝜑) ∧ ∃𝑥(𝑥 = 𝑡 ∧ ¬ 𝜑)), which expresses that there can be no two variables both equal to a given term, one witnessing 𝜑 and the other witnessing ¬ 𝜑. (Contributed by BJ, 21-May-2024.) (Proof modification is discouraged.)

((∃𝑥(𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))))

Theorembj-substw 34076* Weak form of the LHS of bj-subst 34075 proved from the core axiom schemes. Compare ax12w 2138. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.)
(𝑥 = 𝑡 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))

20.15.4.10  Nonfreeness

Syntaxwnnf 34077 Syntax for the nonfreeness quantifier.
wff Ⅎ'𝑥𝜑

Definitiondf-bj-nnf 34078 Definition of the nonfreeness quantifier. The formula Ⅎ'𝑥𝜑 has the intended meaning that the variable 𝑥 is semantically nonfree in the formula 𝜑. The motivation for this quantifier is to have a condition expressible in the logic which is as close as possible to the non-occurrence condition DV (𝑥, 𝜑) (in Metamath files, "\$d x ph \$."), which belongs to the metalogic.

The standard syntactic nonfreeness condition, also expressed in the metalogic, is intermediate between these two notions: semantic nonfreeness implies syntactic nonfreeness, which implies non-occurrence. Both implications are strict; for the first, note that Ⅎ'𝑥𝑥 = 𝑥, that is, 𝑥 is semantically (but not syntactically) nonfree in the formula 𝑥 = 𝑥; for the second, note that 𝑥 is syntactically nonfree in the formula 𝑥𝑥 = 𝑥 although it occurs in it.

We now prove two metatheorems which make precise the above fact that, as far as proving power is concerned, the nonfreeness condition Ⅎ'𝑥𝜑 is very close to the non-occurrence condition DV (𝑥, 𝜑).

Let S be a Metamath system with the FOL-syntax of (i)set.mm, containing intuitionistic positive propositional calculus and ax-5 1912 and ax5e 1914.

1. If the scheme

(Ⅎ'𝑥𝜑 & PHI1 & ... & PHIn PHI0, DV)

is provable in S, then so is the scheme

(PHI1 & ... & PHIn PHI0, DV ∪ {{𝑥, 𝜑}}).

Proof: By bj-nnfv 34105, we can prove (Ⅎ'𝑥𝜑, {{𝑥, 𝜑}}), from which the theorem follows. QED

2. Suppose that S also contains (the FOL version of) modal logic KB and commutation of quantifiers alcom 2164 and excom 2170 (possibly weakened by a DV condition on the quantifying variables), and that S can be axiomatized such that the only axioms with a DV condition involving a formula variable are among ax-5 1912, ax5e 1914, ax5ea 1915. If the scheme

(PHI1 & ... & PHIn PHI0, DV)

is provable in S, then so is the scheme

(Ⅎ'𝑥𝜑 & PHI1 & ... & PHIn PHI0, DV ∖ {{𝑥, 𝜑}}).

More precisely, if S contains modal 45 and if the variables quantified over in PHI0, ..., PHIn are among 𝑥1, ..., 𝑥m, then the scheme

(PHI1 & ... & PHIn (antecedent PHI0), DV ∖ {{𝑥, 𝜑}})

is provable in S, where the antecedent is a finite conjunction of formulas of the form 𝑥i1 ...∀𝑥ip Ⅎ'𝑥𝜑 where the 𝑥ij's are among the 𝑥i's.

Lemma: If 𝑥 OC(PHI), then S proves the scheme

(Ⅎ'𝑥𝜑 ⇒ Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}}).

More precisely, if the variables quantified over in PHI are among 𝑥1, ..., 𝑥m, then

((antecedent → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}})

is provable in S, with the same form of antecedent as above.

Proof: By induction on the height of PHI. We first note that by bj-nnfbi 34079 we can assume that PHI contains only primitive (as opposed to defined) symbols. For the base case, atomic formulas are either 𝜑, in which case the scheme to prove is an instance of id 22, or have variables all in OC(PHI) ∖ {𝜑}, so (Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}}) by bj-nnfv 34105, hence ((Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}}) by a1i 11. For the induction step, PHI is either an implication, a negation, a conjunction, a disjunction, a biconditional, a universal or an existential quantification of formulas where 𝑥 does not occur. We use respectively bj-nnfim 34097, bj-nnfnt 34091, bj-nnfan 34099, bj-nnfor 34101, bj-nnfbit 34103, bj-nnfalt 34117, bj-nnfext 34118. For instance, in the implication case, if we have by induction hypothesis

((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}}) and ((∀𝑦1 ...∀𝑦n Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PSI), {{𝑥, 𝑎} ∣ 𝑎 OC(PSI) ∖ {𝜑}}),

then bj-nnfim 34097 yields

(((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 ∧ ∀𝑦1 ...∀𝑦n Ⅎ'𝑥𝜑) → Ⅎ'𝑥 (PHI PSI)), {{𝑥, 𝑎} ∣ 𝑎 OC(PHI PSI) ∖ {𝜑}})

and similarly for antecedents which are conjunctions as in the statement of the lemma.

In the universal quantification case, say quantification over 𝑦, if we have by induction hypothesis

((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}}),

then bj-nnfalt 34117 yields

((∀𝑦𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥𝑦 PHI), {{𝑥, 𝑎} ∣ 𝑎 OC(𝑦 PHI) ∖ {𝜑}})

and similarly for antecedents which are conjunctions as in the statement of the lemma.

Note bj-nnfalt 34117 and bj-nnfext 34118 are proved from positive propositional calculus with alcom 2164 and excom 2170 (possibly weakened by a DV condition on the quantifying variables), and modalB (via bj-19.12 34112). QED

Proof of the theorem: Consider a proof of that scheme directly from the axioms. Consider a step where a DV condition involving 𝜑 is used. By hypothesis, that step is an instance of ax-5 1912 or ax5e 1914 or ax5ea . It has the form (PSI → ∀𝑥 PSI) where PSI has the form of the lemma and the DV conditions of the proof contain {{𝑥, 𝑎} ∣ 𝑎 OC(PSI) }. Therefore, one has

((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PSI), {{𝑥, 𝑎} ∣ 𝑎 OC(PSI) ∖ {𝜑}})

for appropriate 𝑥i's, and by bj-nnfa 34082 we obtain

((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 (PSI → ∀𝑥 PSI)), {{𝑥, 𝑎} ∣ 𝑎 OC(PSI) ∖ {𝜑}})

and similarly for antecedents which are conjunctions as in the statement of the theorem. Similarly if the step is using ax5e 1914 or ax5ea 1915, we would use bj-nnfe 34084 or bj-nnfea 34086 respectively.

Therefore, taking as antecedent of the theorem to prove the conjunction of all the antecedents at each of these steps, we obtain a proof by "carrying the context over", which is possible, as in the deduction theorem when the step uses ax-mp 5, and when the step uses ax-gen 1797, by bj-nnf-alrim 34106 and bj-nnfa1 34110 (which requires modal 45). The condition DV (𝑥, 𝜑) is not required by the resulting proof.

Finally, there may be in the global antecedent thus constructed some dummy variables, which can be removed by spvw 1986. QED

Compared with df-nf 1786, the present definition is stricter on positive propositional calculus (bj-nnfnfTEMP 34089) and equivalent on core FOL plus sp 2184 (bj-nfnnfTEMP 34109). While being stricter, it still holds for non-occurring variables (bj-nnfv 34105), which is the basic requirement for this quantifier. In particular, it translates more closely the associated variable disjointness condition. Since the nonfreeness quantifier is a means to translate a variable disjointness condition from the metalogic to the logic, it seems preferable. Also, since nonfreeness is mainly used as a hypothesis, this definition would allow more theorems, notably the 19.xx theorems, to be proved from the core axioms, without needing a 19.xxv variant.

One can devise infinitely many definitions increasingly close to the non-occurring condition, like ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ 𝑥((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ ∀𝑥𝑥... and each stronger definition would permit more theorems to be proved from the core axioms. A reasonable rule seems to be to stop before nested quantifiers appear (since they typically require ax-10 2146 to work with), and also not to have redundant conjuncts when full metacomplete FOL= is developed.

(Contributed by BJ, 28-Jul-2023.)

(Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))

Theorembj-nnfbi 34079 If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other. Compare nfbiit 1852. From this and bj-nnfim 34097 and bj-nnfnt 34091, one can prove analogous nonfreeness conservation results for other propositional operators. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 34092) in order not to require sp 2184 (modal T). (Contributed by BJ, 27-Aug-2023.)
(((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))

Theorembj-nnfbd 34080* If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi 34079. (Contributed by BJ, 27-Aug-2023.)
(𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))

Theorembj-nnfbii 34081 If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi 34079. (Contributed by BJ, 18-Nov-2023.)
(𝜑𝜓)       (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)

Theorembj-nnfa 34082 Nonfreeness implies the equivalent of ax-5 1912. See nf5r 2195, nf5ri 2197. (Contributed by BJ, 28-Jul-2023.)
(Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Theorembj-nnfad 34083 Nonfreeness implies the equivalent of ax-5 1912, deduction form. See nf5rd 2198. (Contributed by BJ, 2-Dec-2024.)
(𝜑 → Ⅎ'𝑥𝜓)       (𝜑 → (𝜓 → ∀𝑥𝜓))

Theorembj-nnfe 34084 Nonfreeness implies the equivalent of ax5e 1914. (Contributed by BJ, 28-Jul-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥𝜑𝜑))

Theorembj-nnfed 34085 Nonfreeness implies the equivalent of ax5e 1914, deduction form. (Contributed by BJ, 2-Dec-2024.)
(𝜑 → Ⅎ'𝑥𝜓)       (𝜑 → (∃𝑥𝜓𝜓))

Theorembj-nnfea 34086 Nonfreeness implies the equivalent of ax5ea 1915. (Contributed by BJ, 28-Jul-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))

Theorembj-nnfead 34087 Nonfreeness implies the equivalent of ax5ea 1915, deduction form. (Contributed by BJ, 2-Dec-2024.)
(𝜑 → Ⅎ'𝑥𝜓)       (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Theorembj-dfnnf2 34088 Alternate definition of df-bj-nnf 34078 using only primitive symbols (, ¬, ) in each conjunct. (Contributed by BJ, 20-Aug-2023.)
(Ⅎ'𝑥𝜑 ↔ ((𝜑 → ∀𝑥𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))

Theorembj-nnfnfTEMP 34089 New nonfreeness implies old nonfreeness on implicational calculus (the proof indicates it uses ax-3 8 because of set.mm's definition of the biconditional, but the proof actually holds in minimal implicational calculus). (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1786 except via df-nf 1786 directly. (Proof modification is discouraged.)
(Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)

Theorembj-wnfnf 34090 When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). Note that this could also be proved from bj-nnfim 34097, bj-nnfe1 34111 and bj-nnfa1 34110. (Contributed by BJ, 9-Dec-2023.)
Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)

Theorembj-nnfnt 34091 A variable is nonfree in a formula if and only if it is nonfree in its negation. The foward implication is intuitionistically valid (and that direction is sufficient for the purpose of recursively proving that some formulas have a given variable not free in them, like bj-nnfim 34097). Intuitionistically, (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1857. (Contributed by BJ, 28-Jul-2023.)
(Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)

Theorembj-nnftht 34092 A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2184 (modal T), as in bj-nnfbi 34079. (Contributed by BJ, 28-Jul-2023.)
((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)

Theorembj-nnfth 34093 A variable is nonfree in a theorem, inference form. (Contributed by BJ, 28-Jul-2023.)
𝜑       Ⅎ'𝑥𝜑

Theorembj-nnfnth 34094 A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.)
¬ 𝜑       Ⅎ'𝑥𝜑

Theorembj-nnfim1 34095 A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))

Theorembj-nnfim2 34096 A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑𝜓)))

Theorembj-nnfim 34097 Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication. (Contributed by BJ, 27-Aug-2023.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))

Theorembj-nnfimd 34098 Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication, deduction form. (Contributed by BJ, 2-Dec-2023.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))

Theorembj-nnfan 34099 Nonfreeness in both conjuncts implies nonfreeness in the conjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of conjunction in terms of implication and negation, so using bj-nnfim 34097, bj-nnfnt 34091 and bj-nnfbi 34079, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))

Theorembj-nnfand 34100 Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 34099, it has two more essential steps but fewer total steps (since there are fewer intermediate formulas to build) and is easier to follow and understand. This statement is of intermediate complexity: for simpler statements, closed-style proofs like that of bj-nnfan 34099 will generally be shorter than deduction-style proofs while still easy to follow, while for more complex statements, the opposite will be true (and deduction-style proofs like that of bj-nnfand 34100 will generally be easier to understand). (Contributed by BJ, 19-Nov-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 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