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Theorem List for Metamath Proof Explorer - 33901-34000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-spnfw 33901 Theorem close to a closed form of spnfw 1975. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) → (∀𝑥𝜑𝜓))
 
Theorembj-cbvexiw 33902* Change bound variable. This is to cbvexvw 2035 what cbvaliw 2004 is to cbvalvw 2034. TODO: move after cbvalivw 2005. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)
 
Theorembj-cbvexivw 33903* Change bound variable. This is to cbvexvw 2035 what cbvalivw 2005 is to cbvalvw 2034. TODO: move after cbvalivw 2005. (Contributed by BJ, 17-Mar-2020.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥𝜑 → ∃𝑦𝜓)
 
Theorembj-modald 33904 A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
 
Theorembj-denot 33905* A weakening of ax-6 1961 and ax6v 1962. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
(𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)
 
Theorembj-eqs 33906* A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2383. 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))
 
20.15.4.6  Adding ax-7
 
Theorembj-cbvexw 33907* Change bound variable. This is to cbvexvw 2035 what cbvalw 2033 is to cbvalvw 2034. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝑦𝜓 → ∃𝑦𝜓)    &   (𝜑 → ∀𝑦𝜑)    &   (∃𝑦𝑥𝜑 → ∃𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
Theorembj-ax12w 33908* The general statement that ax12w 2128 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 33909 A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2107 and ax-9 2115. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2107 and ax-9 2115, as proved here. In the other direction, one can prove ax-8 2107 (respectively ax-9 2115) from bj-ax89 33909 by using mpan2 687 (respectively mpan 686) and equid 2010. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))
 
Theorembj-elequ12 33910 An identity law for the non-logical predicate, which combines elequ1 2112 and elequ2 2120. For the analogous theorems for class terms, see eleq1 2900, eleq2 2901 and eleq12 2902. TODO: move to main part. (Contributed by BJ, 29-Sep-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))
 
Theorembj-cleljusti 33911* One direction of cleljust 2114, requiring only ax-1 6-- ax-5 1902 and ax8v1 2109. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
(∃𝑧(𝑧 = 𝑥𝑧𝑦) → 𝑥𝑦)
 
20.15.4.8  Adding ax-11
 
Theorembj-alcomexcom 33912 Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1801 section, soon after 2nexaln 1821, and used to prove excom 2159. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))
 
Theorembj-hbalt 33913 Closed form of hbal 2164. When in main part, prove hbal 2164 and hbald 2165 from it. (Contributed by BJ, 2-May-2019.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))
 
20.15.4.9  Adding ax-12
 
Theoremaxc11n11 33914 Proof of axc11n 2443 from { ax-1 6-- ax-7 2006, axc11 2447 } . Almost identical to axc11nfromc11 35944. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Theoremaxc11n11r 33915 Proof of axc11n 2443 from { ax-1 6-- ax-7 2006, axc9 2393, axc11r 2379 } (note that axc16 2253 is provable from { ax-1 6-- ax-7 2006, axc11r 2379 }).

Note that axc11n 2443 proves (over minimal calculus) that axc11 2447 and axc11r 2379 are equivalent. Therefore, axc11n11 33914 and axc11n11r 33915 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2447 appears slightly stronger since axc11n11r 33915 requires axc9 2393 while axc11n11 33914 does not).

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

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Theorembj-axc16g16 33916* Proof of axc16g 2252 from { ax-1 6-- ax-7 2006, axc16 2253 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
 
Theorembj-ax12v3 33917* A weak version of ax-12 2167 which is stronger than ax12v 2168. Note that if one assumes reflexivity of equality 𝑥 = 𝑥 (equid 2010), then bj-ax12v3 33917 implies ax-5 1902 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 33918. (Contributed by BJ, 6-Jul-2021.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-ax12v3ALT 33918* Alternate proof of bj-ax12v3 33917. Uses axc11r 2379 and axc15 2438 instead of ax-12 2167. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-sb 33919* A weak variant of sbid2 2546 not requiring ax-13 2383 nor ax-10 2136. On top of Tarski's FOL, one implication requires only ax12v 2168, and the other requires only sp 2172. (Contributed by BJ, 25-May-2021.)
(𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorembj-modalbe 33920 The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2330. (Contributed by BJ, 20-Oct-2019.)
(𝜑 → ∀𝑥𝑥𝜑)
 
Theorembj-spst 33921 Closed form of sps 2174. Once in main part, prove sps 2174 and spsd 2176 from it. (Contributed by BJ, 20-Oct-2019.)
((𝜑𝜓) → (∀𝑥𝜑𝜓))
 
Theorembj-19.21bit 33922 Closed form of 19.21bi 2178. (Contributed by BJ, 20-Oct-2019.)
((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
 
Theorembj-19.23bit 33923 Closed form of 19.23bi 2180. (Contributed by BJ, 20-Oct-2019.)
((∃𝑥𝜑𝜓) → (𝜑𝜓))
 
Theorembj-nexrt 33924 Closed form of nexr 2181. Contrapositive of 19.8a 2170. (Contributed by BJ, 20-Oct-2019.)
(¬ ∃𝑥𝜑 → ¬ 𝜑)
 
Theorembj-alrim 33925 Closed form of alrimi 2204. (Contributed by BJ, 2-May-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
 
Theorembj-alrim2 33926 Uncurried (imported) form of bj-alrim 33925. (Contributed by BJ, 2-May-2019.)
((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (𝜑 → ∀𝑥𝜓))
 
Theorembj-nfdt0 33927 A theorem close to a closed form of nf5d 2284 and nf5dh 2142. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))
 
Theorembj-nfdt 33928 Closed form of nf5d 2284 and nf5dh 2142. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
 
Theorembj-nexdt 33929 Closed form of nexd 2214. (Contributed by BJ, 20-Oct-2019.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
 
Theorembj-nexdvt 33930* Closed form of nexdv 1928. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
 
Theorembj-alexbiex 33931 Adding a second quantifier is a tranparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-exexbiex 33932 Adding a second quantifier is a tranparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-alalbial 33933 Adding a second quantifier is a tranparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.)
(∀𝑥𝑥𝜑 ↔ ∀𝑥𝜑)
 
Theorembj-exalbial 33934 Adding a second quantifier is a tranparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)
(∃𝑥𝑥𝜑 ↔ ∀𝑥𝜑)
 
Theorembj-19.9htbi 33935 Strengthening 19.9ht 2331 by replacing its succedent with a biconditional (19.9t 2195 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
 
Theorembj-hbntbi 33936 Strengthening hbnt 2294 by replacing its succedent with a biconditional. See also hbntg 32948 and hbntal 40767. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 33935. (Proof modification is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
 
Theorembj-biexal1 33937 A general FOL biconditional that generalizes 19.9ht 2331 among others. For this and the following theorems, see also 19.35 1869, 19.21 2198, 19.23 2202. 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 33938 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 33939 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 33940 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 33941 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 33942 Closed form of hbex 2336. (Contributed by BJ, 10-Oct-2019.)
(∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑))
 
Theorembj-nfalt 33943 Closed form of nfal 2334. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
 
Theorembj-nfext 33944 Closed form of nfex 2335. (Contributed by BJ, 10-Oct-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
 
Theorembj-eeanvw 33945* Version of exdistrv 1947 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2151. (The same can be done with eeeanv 2363 and ee4anv 2364.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 
Theorembj-modal4 33946 First-order logic form of the modal axiom (4). See hba1 2293. This is the standard proof of the implication in modal logic (B5 4). Its dual statement is bj-modal4e 33947. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)
 
Theorembj-modal4e 33947 First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 33946 (hba1 2293). (Contributed by BJ, 21-Dec-2020.) (Proof modification is discouraged.)
(∃𝑥𝑥𝜑 → ∃𝑥𝜑)
 
Theorembj-modalb 33948 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 33949 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 33950 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 33951 When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the "forall" form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
 
Theorembj-wnfenf 33952 When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the "exists" form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑𝜓))
 
20.15.4.10  Nonfreeness
 
Syntaxwnnf 33953 Syntax for the nonfreeness quantifier.
wff Ⅎ'𝑥𝜑
 
Definitiondf-bj-nnf 33954 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 1902 and ax5e 1904.

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 33981, 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 2153 and excom 2159 (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 1902, ax5e 1904, ax5ea 1905. 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 33955 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 33981, 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 33973, bj-nnfnt 33967, bj-nnfan 33975, bj-nnfor 33977, bj-nnfbit 33979, bj-nnfalt 33993, bj-nnfext 33994. 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 33973 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 33993 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 33993 and bj-nnfext 33994 are proved from positive propositional calculus with alcom 2153 and excom 2159 (possibly weakened by a DV condition on the quantifying variables), and modalB (via bj-19.12 33988). 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 1902 or ax5e 1904 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 33958 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 1904 or ax5ea 1905, we would use bj-nnfe 33960 or bj-nnfea 33962 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 1787, by bj-nnf-alrim 33982 and bj-nnfa1 33986 (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 1976. QED

Compared with df-nf 1776, the present definition is stricter on positive propositional calculus (bj-nnfnfTEMP 33965) and equivalent on core FOL plus sp 2172 (bj-nfnnfTEMP 33985). While being stricter, it still holds for non-occurring variables (bj-nnfv 33981), 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 2136 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 33955 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 1842. From this and bj-nnfim 33973 and bj-nnfnt 33967, 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 33968) in order not to require sp 2172 (modal T). (Contributed by BJ, 27-Aug-2023.)
(((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))
 
Theorembj-nnfbd 33956* 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 33955. (Contributed by BJ, 27-Aug-2023.)
(𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
 
Theorembj-nnfbii 33957 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 33955. (Contributed by BJ, 18-Nov-2023.)
(𝜑𝜓)       (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)
 
Theorembj-nnfa 33958 Nonfreeness implies the equivalent of ax-5 1902. See nf5r 2183, nf5ri 2185. (Contributed by BJ, 28-Jul-2023.)
(Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 
Theorembj-nnfad 33959 Nonfreeness implies the equivalent of ax-5 1902, deduction form. See nf5rd 2187. (Contributed by BJ, 2-Dec-2024.)
(𝜑 → Ⅎ'𝑥𝜓)       (𝜑 → (𝜓 → ∀𝑥𝜓))
 
Theorembj-nnfe 33960 Nonfreeness implies the equivalent of ax5e 1904. (Contributed by BJ, 28-Jul-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥𝜑𝜑))
 
Theorembj-nnfed 33961 Nonfreeness implies the equivalent of ax5e 1904, deduction form. (Contributed by BJ, 2-Dec-2024.)
(𝜑 → Ⅎ'𝑥𝜓)       (𝜑 → (∃𝑥𝜓𝜓))
 
Theorembj-nnfea 33962 Nonfreeness implies the equivalent of ax5ea 1905. (Contributed by BJ, 28-Jul-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
 
Theorembj-nnfead 33963 Nonfreeness implies the equivalent of ax5ea 1905, deduction form. (Contributed by BJ, 2-Dec-2024.)
(𝜑 → Ⅎ'𝑥𝜓)       (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
 
Theorembj-dfnnf2 33964 Alternate definition of df-bj-nnf 33954 using only primitive symbols (, ¬, ). (Contributed by BJ, 20-Aug-2023.)
(Ⅎ'𝑥𝜑 ↔ ((𝜑 → ∀𝑥𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
 
Theorembj-nnfnfTEMP 33965 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 1776 except via df-nf 1776 directly. (Proof modification is discouraged.)
(Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)
 
Theorembj-wnfnf 33966 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 33973, bj-nnfe1 33987 and bj-nnfa1 33986. (Contributed by BJ, 9-Dec-2023.)
Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
 
Theorembj-nnfnt 33967 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 33973). Intuitionistically, (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1847. (Contributed by BJ, 28-Jul-2023.)
(Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)
 
Theorembj-nnftht 33968 A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2172 (modal T), as in bj-nnfbi 33955. (Contributed by BJ, 28-Jul-2023.)
((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)
 
Theorembj-nnfth 33969 A variable is nonfree in a theorem, inference form. (Contributed by BJ, 28-Jul-2023.)
𝜑       Ⅎ'𝑥𝜑
 
Theorembj-nnfnth 33970 A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.)
¬ 𝜑       Ⅎ'𝑥𝜑
 
Theorembj-nnfim1 33971 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 33972 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 33973 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 33974 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 33975 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 33973, bj-nnfnt 33967 and bj-nnfbi 33955, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))
 
Theorembj-nnfand 33976 Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 33975, 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 33975 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 33976 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))
 
Theorembj-nnfor 33977 Nonfreeness in both disjuncts implies nonfreeness in the disjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of disjunction in terms of implication and negation, so using bj-nnfim 33973, bj-nnfnt 33967 and bj-nnfbi 33955, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))
 
Theorembj-nnford 33978 Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 33977 and bj-nnfand 33976. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))
 
Theorembj-nnfbit 33979 Nonfreeness in both sides implies nonfreeness in the biconditional. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))
 
Theorembj-nnfbid 33980 Nonfreeness in both sides implies nonfreeness in the biconditional, deduction form. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))
 
Theorembj-nnfv 33981* A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.)
Ⅎ'𝑥𝜑
 
Theorembj-nnf-alrim 33982 Proof of the closed form of alrimi 2204 from modalK (compare alrimiv 1919). See also bj-alrim 33925. Actually, most proofs between 19.3t 2192 and 2sbbid 2238 could be proved without ax-12 2167. (Contributed by BJ, 20-Aug-2023.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
 
Theorembj-nnf-exlim 33983 Proof of the closed form of exlimi 2208 from modalK (compare exlimiv 1922). See also bj-sylget2 33853. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓)))
 
Theorembj-dfnnf3 33984 Alternate definition of nonfreeness when sp 2172 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1776. (Proof modification is discouraged.)
(Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
 
Theorembj-nfnnfTEMP 33985 New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2172. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1776 except via df-nf 1776 directly. (Proof modification is discouraged.)
(Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)
 
Theorembj-nnfa1 33986 See nfa1 2146. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
Ⅎ'𝑥𝑥𝜑
 
Theorembj-nnfe1 33987 See nfe1 2145. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
Ⅎ'𝑥𝑥𝜑
 
Theorembj-19.12 33988 See 19.12 2338. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2159 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1776 or df-bj-nnf 33954, directly or indirectly. (Proof modification is discouraged.)
(∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theorembj-nnflemaa 33989 One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 33913. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝑥𝜑))
 
Theorembj-nnflemee 33990 One of four lemmas for nonfreeness: antecedent and consequent both expressed using existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥(∃𝑦𝜑𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝜑))
 
Theorembj-nnflemae 33991 One of four lemmas for nonfreeness: antecedent expressed with universal quantifier and consequent expressed with existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦𝑥𝜑))
 
Theorembj-nnflemea 33992 One of four lemmas for nonfreeness: antecedent expressed with existential quantifier and consequent expressed with universal quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥(∃𝑦𝜑𝜑) → (∃𝑦𝑥𝜑 → ∀𝑥𝜑))
 
Theorembj-nnfalt 33993 See nfal 2334 and bj-nfalt 33943. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦𝑥𝜑)
 
Theorembj-nnfext 33994 See nfex 2335 and bj-nfext 33944. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦𝑥𝜑)
 
Theorembj-stdpc5t 33995 Alias of bj-nnf-alrim 33982 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2199 proved from modalK (obsoleting stdpc5v 1930). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 33982 instead. (New usaged is discouraged.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
 
Theorembj-19.21t 33996 Statement 19.21t 2197 proved from modalK (obsoleting 19.21v 1931). (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 
Theorembj-19.23t 33997 Statement 19.23t 2201 proved from modalK (obsoleting 19.23v 1934). (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
 
Theorembj-19.36im 33998 One direction of 19.36 2223 from the same axioms as 19.36imv 1937. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓)))
 
Theorembj-19.37im 33999 One direction of 19.37 2225 from the same axioms as 19.37imv 1939. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓)))
 
Theorembj-19.42t 34000 Closed form of 19.42 2229 from the same axioms as 19.42v 1945. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))
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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 44801-44804
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