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Theorem List for Metamath Proof Explorer - 32901-33000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-abbi2i 32901* Remove dependency on ax-13 2282 from abbi2i 2767. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}
 
Theorembj-abbii 32902 Remove dependency on ax-13 2282 from abbii 2768. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}
 
Theorembj-abbid 32903 Remove dependency on ax-13 2282 from abbid 2769. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 
Theorembj-abbidv 32904* Remove dependency on ax-13 2282 from abbidv 2770. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 
Theorembj-abbi2dv 32905* Remove dependency on ax-13 2282 from abbi2dv 2771. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})
 
Theorembj-abbi1dv 32906* Remove dependency on ax-13 2282 from abbi1dv 2772. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} = 𝐴)
 
Theorembj-abid2 32907* Remove dependency on ax-13 2282 from abid2 2774. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
{𝑥𝑥𝐴} = 𝐴
 
Theorembj-clabel 32908* Remove dependency on ax-13 2282 from clabel 2778 (note the absence of DV conditions among variables in the LHS). (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
 
Theorembj-sbab 32909* Remove dependency on ax-13 2282 from sbab 2779 (note the absence of DV conditions among variables in the LHS). (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧𝐴})
 
Theorembj-nfab1 32910 Remove dependency on ax-13 2282 from nfab1 2795 (note the absence of DV conditions). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥{𝑥𝜑}
 
Theorembj-vjust 32911 Remove dependency on ax-13 2282 from vjust 3232 (note the absence of DV conditions). Soundness justification theorem for df-v 3233. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
{𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
 
Theorembj-cdeqab 32912* Remove dependency on ax-13 2282 from cdeqab 3458. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})
 
Theorembj-axrep1 32913* Remove dependency on ax-13 2282 from axrep1 4805. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))
 
Theorembj-axrep2 32914* Remove dependency on ax-13 2282 from axrep2 4806. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
 
Theorembj-axrep3 32915* Remove dependency on ax-13 2282 from axrep3 4807. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
 
Theorembj-axrep4 32916* Remove dependency on ax-13 2282 from axrep4 4808. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑧𝜑       (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
 
Theorembj-axrep5 32917* Remove dependency on ax-13 2282 from axrep5 4809. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑧𝜑       (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
 
Theorembj-axsep 32918* Remove dependency on ax-13 2282 from axsep 4813. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
Theorembj-nalset 32919* Remove dependency on ax-13 2282 from nalset 4828. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
¬ ∃𝑥𝑦 𝑦𝑥
 
Theorembj-zfpow 32920* Remove dependency on ax-13 2282 from zfpow 4874. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)
 
Theorembj-el 32921* Remove dependency on ax-13 2282 from el 4877. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑦 𝑥𝑦
 
Theorembj-dtru 32922* Remove dependency on ax-13 2282 from dtru 4887. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
¬ ∀𝑥 𝑥 = 𝑦
 
Theorembj-axc16b 32923* Remove dependency on ax-13 2282 from axc16b 4888. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 
Theorembj-eunex 32924 Remove dependency on ax-13 2282 from eunex 4889. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
 
Theorembj-dtrucor 32925* Remove dependency on ax-13 2282 from dtrucor 4930. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥 = 𝑦       𝑥𝑦
 
Theorembj-dtrucor2v 32926* Version of dtrucor2 4931 with a dv condition, which does not require ax-13 2282 (nor ax-4 1777, ax-5 1879, ax-7 1981, ax-12 2087). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦𝑥𝑦)       (𝜑 ∧ ¬ 𝜑)
 
Theorembj-dvdemo1 32927* Remove dependency on ax-13 2282 from dvdemo1 4932 (this removal is noteworthy since dvdemo1 4932 and dvdemo2 4933 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑦𝑧𝑥)
 
Theorembj-dvdemo2 32928* Remove dependency on ax-13 2282 from dvdemo2 4933 (this removal is noteworthy since dvdemo1 4932 and dvdemo2 4933 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑦𝑧𝑥)
 
20.14.4.12  Strengthenings of theorems of the main part

Typically, these are biconditional versions of theorems in the main part which are formulated as implications. They could be added after said implication, or sometimes replace it (by "inlining" it).

This could also be done for hba1 2189, hbe1 2061, hbn1 2060, modal-5 2072.

 
Theorembj-sb3b 32929 Simplified definition of substitution when variables are distinct. This is to sb3 2383 what sb4b 2386 is to sb4 2384. Actually, one may keep only bj-sb3b 32929 and sb4b 2386 in the database, renaming them sb3 and sb4. (Contributed by BJ, 6-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
 
20.14.4.13  Distinct var metavariables

The closed formula 𝑥𝑦𝑥 = 𝑦 approximately means that the var metavariables 𝑥 and 𝑦 represent the same variable vi. In a domain with at most one object, however, this formula is always true, hence the "approximately" in the previous sentence.

 
Theorembj-hbaeb2 32930 Biconditional version of a form of hbae 2348 with commuted quantifiers, not requiring ax-11 2074. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑧 𝑥 = 𝑦)
 
Theorembj-hbaeb 32931 Biconditional version of hbae 2348. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧𝑥 𝑥 = 𝑦)
 
Theorembj-hbnaeb 32932 Biconditional version of hbnae 2350 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
 
Theorembj-dvv 32933 A special instance of bj-hbaeb2 32930. A lemma for distinct var metavariables. Note that the right-hand side is a closed formula (a sentence). (Contributed by BJ, 6-Oct-2018.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑦 𝑥 = 𝑦)
 
20.14.4.14  Around ~ equsal

As a rule of thumb, if a theorem of the form (𝜑𝜓) ⇒ (𝜒𝜃) is in the database, and the "more precise" theorems (𝜑𝜓) ⇒ (𝜒𝜃) and (𝜓𝜑) ⇒ (𝜃𝜒) also hold (see bj-bisym 32700), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2328 (and equsalh 2330 and equsexh 2331). Even if only one of these two theorems holds, it should be added to the database.

 
Theorembj-equsal1t 32934 Duplication of wl-equsal1t 33457, with shorter proof. If one imposes a DV condition on x,y , then one can use bj-alequexv 32780 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 33458 is also interesting. (Contributed by BJ, 6-Oct-2018.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
 
Theorembj-equsal1ti 32935 Inference associated with bj-equsal1t 32934. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)
 
Theorembj-equsal1 32936 One direction of equsal 2327. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) → 𝜓)
 
Theorembj-equsal2 32937 One direction of equsal 2327. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓))
 
Theorembj-equsal 32938 Shorter proof of equsal 2327. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2327, but "min */exc equsal" is ok. (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
20.14.4.15  Some Principia Mathematica proofs

References are made to the second edition (1927, reprinted 1963) of Principia Mathematica, Vol. 1. Theorems are referred to in the form "PM*xx.xx".

 
Theoremstdpc5t 32939 Closed form of stdpc5 2114. (Possible to place it before 19.21t 2111 and use it to prove 19.21t 2111). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
 
Theorembj-stdpc5 32940 More direct proof of stdpc5 2114. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
 
Theorem2stdpc5 32941 A double stdpc5 2114 (one direction of PM*11.3). See also 2stdpc4 2382 and 19.21vv 38892. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) → (𝜑 → ∀𝑥𝑦𝜓))
 
Theorembj-19.21t 32942 Proof of 19.21t 2111 from stdpc5t 32939. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 
Theoremexlimii 32943 Inference associated with exlimi 2124. Inferring a theorem when it is implied by an antecedent which may be true. (Contributed by BJ, 15-Sep-2018.)
𝑥𝜓    &   (𝜑𝜓)    &   𝑥𝜑       𝜓
 
Theoremax11-pm 32944 Proof of ax-11 2074 similar to PM's proof of alcom 2077 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 32948. Axiom ax-11 2074 is used in the proof only through nfa2 2080. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremax6er 32945 Commuted form of ax6e 2286. (Could be placed right after ax6e 2286). (Contributed by BJ, 15-Sep-2018.)
𝑥 𝑦 = 𝑥
 
Theoremexlimiieq1 32946 Inferring a theorem when it is implied by an equality which may be true. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑    &   (𝑥 = 𝑦𝜑)       𝜑
 
Theoremexlimiieq2 32947 Inferring a theorem when it is implied by an equality which may be true. (Contributed by BJ, 15-Sep-2018.) (Revised by BJ, 30-Sep-2018.)
𝑦𝜑    &   (𝑥 = 𝑦𝜑)       𝜑
 
Theoremax11-pm2 32948* Proof of ax-11 2074 from the standard axioms of predicate calculus, similar to PM's proof of alcom 2077 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 2074 is used in the proof only through nfal 2191, nfsb 2468, sbal 2490, sb8 2452. See also ax11-pm 32944. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
20.14.4.16  Alternate definition of substitution
 
Theorembj-sbsb 32949 Biconditional showing two possible (dual) definitions of substitution df-sb 1938 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
(((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜑)))
 
Theorembj-dfsb2 32950 Alternate (dual) definition of substitution df-sb 1938 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
([𝑦 / 𝑥]𝜑 ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜑)))
 
20.14.4.17  Lemmas for substitution
 
Theorembj-sbf3 32951 Substitution has no effect on a bound variabe (existential quantifier case); see sbf2 2410. (Contributed by BJ, 2-May-2019.)
([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-sbf4 32952 Substitution has no effect on a bound variabe (non-freeness case); see sbf2 2410. (Contributed by BJ, 2-May-2019.)
([𝑦 / 𝑥]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜑)
 
Theorembj-sbnf 32953* Move non-free predicate in and out of substitution; see sbal 2490 and sbex 2491. (Contributed by BJ, 2-May-2019.)
([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)
 
20.14.4.18  Existential uniqueness
 
Theorembj-eu3f 32954* Version of eu3v 2526 where the dv condition is replaced with a non-freeness hypothesis. This is a "backup" of a theorem that used to be in the main part with label "eu3" and was deprecated in favor of eu3v 2526. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)
𝑦𝜑       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theorembj-eumo0 32955* Existential uniqueness implies "at most one." Used to be in the main part and deprecated in favor of eumo 2527 and mo2 2507. (Contributed by NM, 8-Jul-1994.) (Revised by BJ, 8-Jun-2019.)
𝑦𝜑       (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
20.14.4.19  First-logic: miscellaneous

Miscellaneous theorems of first-order logic.

 
Theorembj-sbidmOLD 32956 Obsolete proof of sbidm 2442 temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 
Theorembj-mo3OLD 32957* Obsolete proof of mo3 2536 temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
 
Theorembj-syl66ib 32958 A mixed syllogism inference derived from syl6ib 241. In addition to bj-dvelimdv1 32960, it can also shorten alexsubALTlem4 21901 (4821>4812), supsrlem 9970 (2868>2863). (Contributed by BJ, 20-Oct-2021.)
(𝜑 → (𝜓𝜃))    &   (𝜃𝜏)    &   (𝜏𝜒)       (𝜑 → (𝜓𝜒))
 
Theorembj-dvelimdv 32959* Deduction form of dvelim 2368 with DV conditions. Uncurried (imported) form of bj-dvelimdv 32959. Typically, 𝑧 is a fresh variable used for the implicit substitution hypothesis that results in 𝜒 (namely, 𝜓 can be thought as 𝜓(𝑥, 𝑦) and 𝜒 as 𝜓(𝑥, 𝑧)). So the theorem says that if x is effectively free in 𝜓(𝑥, 𝑧), then if x and y are not the same variable, then 𝑥 is also effectively free in 𝜓(𝑥, 𝑦), in a context 𝜑.

One can weakend the implicit substitution hypothesis by adding the antecedent 𝜑 but this typically does not make the theorem much more useful. Similarly, one could use non-freeness hypotheses instead of DV conditions but since this result is typically used when 𝑧 is a dummy variable, this would not be of much benefit. One could also remove DV(z,x) since in the proof nfv 1883 can be replaced with nfal 2191 followed by nfn 1824.

Remark: nfald 2201 uses ax-11 2074; it might be possible to inline and use ax11w 2047 instead, but there is still a use via 19.12 2200 anyway. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝑧 = 𝑦 → (𝜒𝜓))       ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
 
Theorembj-dvelimdv1 32960* Curried (exported) form of bj-dvelimdv 32959. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝑧 = 𝑦 → (𝜒𝜓))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓))
 
Theorembj-dvelimv 32961* A version of dvelim 2368 using the "non-free" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑧 = 𝑦 → (𝜓𝜑))       (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
 
Theorembj-nfeel2 32962* Non-freeness in an equality. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝑧)
 
Theorembj-axc14nf 32963 Proof of a version of axc14 2400 using the "non-free" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥𝑦))
 
Theorembj-axc14 32964 Alternate proof of axc14 2400 (even when inlining the above results, this gives a shorter proof). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
 
20.14.5  Set theory
 
20.14.5.1  Eliminability of class terms

In this section, we give a sketch of the proof of the Eliminability Theorem for class terms in an extensional set theory where quantification occurs only over set variables.

Eliminability of class variables using the $a-statements ax-ext 2631, df-clab 2638, df-cleq 2644, df-clel 2647 is an easy result, proved for instance in Appendix X of Azriel Levy, Basic Set Theory, Dover Publications, 2002. Note that viewed from the set.mm axiomatization, it is a metatheorem not formalizable is set.mm. It states: every formula in the language of FOL + + class terms, but without class variables, is provably equivalent (over {FOL, ax-ext 2631, df-clab 2638, df-cleq 2644, df-clel 2647 }) to a formula in the language of FOL + (that is, without class terms).

The proof goes by induction on the complexity of the formula (see op. cit. for details). The base case is that of atomic formulas. The atomic formulas containing class terms are of one of the following forms: for equality, 𝑥 = {𝑦𝜑}, {𝑥𝜑} = 𝑦, {𝑥𝜑} = {𝑦𝜓}, and for membership, 𝑦 ∈ {𝑥𝜑}, {𝑥𝜑} ∈ 𝑦, {𝑥𝜑} ∈ {𝑦𝜓}. These cases are dealt with by eliminable1 32965 and the following theorems of this section, which are special instances of df-clab 2638, dfcleq 2645 (proved from {FOL, ax-ext 2631, df-cleq 2644 }), and df-clel 2647. Indeed, denote by (i) the formula proved by "eliminablei". One sees that the RHS of (1) has no class terms, the RHS's of (2x) have only class terms of the form dealt with by (1), and the RHS's of (3x) have only class terms of the forms dealt with by (1) and (2a). Note that in order to prove eliminable2a 32966, eliminable2b 32967 and eliminable3a 32969, we need to substitute a class variable for a setvar variable. This is possible because setvars are class terms: this is the content of the syntactic theorem cv 1522, which is used in these proofs (this does not appear in the html pages but it is in the set.mm file and you can check it using the Metamath program).

The induction step relies on the fact that any formula is a FOL-combination of atomic formulas, so if one found equivalents for all atomic formulas constituting the formula, then the same FOL-combination of these equivalents will be equivalent to the original formula.

Note that one has a slightly more precise result: if the original formula has only class terms appearing in atomic formulas of the form 𝑦 ∈ {𝑥𝜑}, then df-clab 2638 is sufficient (over FOL) to eliminate class terms, and if the original formula has only class terms appearing in atomic formulas of the form 𝑦 ∈ {𝑥𝜑} and equalities, then df-clab 2638, ax-ext 2631 and df-cleq 2644 are sufficient (over FOL) to eliminate class terms.

To prove that { df-clab 2638, df-cleq 2644, df-clel 2647 } provides a definitional extension of {FOL, ax-ext 2631 }, one needs to prove the above Eliminability Theorem, which compares the expressive powers of the languages with and without class terms, and the Conservativity Theorem, which compares the deductive powers when one adds { df-clab 2638, df-cleq 2644, df-clel 2647 }. It states that a formula without class terms is provable in one axiom system if and only if it is provable in the other, and that this remains true when one adds further definitions to {FOL, ax-ext 2631 }. It is also proved in op. cit. The proof is more difficult, since one has to construct for each proof of a statement without class terms, an associated proof not using { df-clab 2638, df-cleq 2644, df-clel 2647 }. It involves a careful case study on the structure of the proof tree.

 
Theoremeliminable1 32965 A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
 
Theoremeliminable2a 32966* A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = {𝑦𝜑} ↔ ∀𝑧(𝑧𝑥𝑧 ∈ {𝑦𝜑}))
 
Theoremeliminable2b 32967* A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝜑} = 𝑦 ↔ ∀𝑧(𝑧 ∈ {𝑥𝜑} ↔ 𝑧𝑦))
 
Theoremeliminable2c 32968* A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝜑} = {𝑦𝜓} ↔ ∀𝑧(𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓}))
 
Theoremeliminable3a 32969* A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝜑} ∈ 𝑦 ↔ ∃𝑧(𝑧 = {𝑥𝜑} ∧ 𝑧𝑦))
 
Theoremeliminable3b 32970* A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝜑} ∈ {𝑦𝜓} ↔ ∃𝑧(𝑧 = {𝑥𝜑} ∧ 𝑧 ∈ {𝑦𝜓}))
 
Theorembj-termab 32971* Every class can be written as (is equal to) a class abstraction. cvjust 2646 is a special instance of it, but the present proof does not require ax-13 2282, contrary to cvjust 2646. This theorem requires ax-ext 2631, df-clab 2638, df-cleq 2644, df-clel 2647, but to prove that any specific class term not containing class variables is a setvar or can be written as (is equal to) a class abstraction does not require these $a-statements. This last fact is a metatheorem, consequence of the fact that the only $a-statements with typecode class are cv 1522, cab 2637 and statements corresponding to defined class constructors.

UPDATE: This theorem is (almost) abid2 2774 and bj-abid2 32907, though the present proof is shorter than a proof from bj-abid2 32907 and eqcomi 2660 (and is shorter than the proof of either); plus, it is of the same form as cvjust 2646 and such a basic statement deserves to be present in both forms. Note that bj-termab 32971 shortens the proof of abid2 2774, and shortens five proofs by a total of 72 bytes. Move it to Main as "abid1" proved from abbi2i 2767? Note also that this is the form in Quine, more than abid2 2774. (Contributed by BJ, 21-Oct-2019.) (Proof modification is discouraged.)

𝐴 = {𝑥𝑥𝐴}
 
20.14.5.2  Classes without extensionality

A few results about classes can be proved without using ax-ext 2631. One could move all theorems from cab 2637 to df-clel 2647 (except for dfcleq 2645 and cvjust 2646) in a subsection "Classes" before the subsection on the axiom of extensionality, together with the theorems below. In that subsection, the last statement should be df-cleq 2644.

Note that without ax-ext 2631, the $a-statements df-clab 2638, df-cleq 2644, and df-clel 2647 are no longer eliminable (see previous section) (but PROBABLY are still conservative). This is not a reason not to study what is provable with them but without ax-ext 2631, in order to gauge their strengths more precisely.

Before that subsection, a subsection "The membership predicate" could group the statements with that are currently in the FOL part (including wcel 2030, wel 2031, ax-8 2032, ax-9 2039).

Remark: the weakening of eleq1 2718 / eleq2 2719 to eleq1w 2713 / eleq2w 2714 can also be done with eleq1i 2721, eqeltri 2726, eqeltrri 2727, eleq1a 2725, eleq1d 2715, eqeltrd 2730, eqeltrrd 2731, eqneltrd 2749, eqneltrrd 2750, nelneq 2754.

 
Theorembj-cleljustab 32972* An instance of df-clel 2647 where the LHS (the definiendum) has the form "setvar class abstraction". The straightforward yet important fact that this statement can be proved from FOL= and df-clab 2638 (hence without df-clel 2647 or df-cleq 2644) was stressed by Mario Carneiro. The instance of df-clel 2647 where the LHS has the form "setvar setvar" is proved as cleljust 2038, from FOL= and ax-8 2032. Note: when df-ssb 32745 is the official definition for substitution, one can use bj-ssbequ instead of sbequ 2404 to prove bj-cleljustab 32972 from Tarski's FOL= with df-clab 2638. (Contributed by BJ, 8-Nov-2021.) (Proof modification is discouraged.)
(𝑥 ∈ {𝑦𝜑} ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
 
Theorembj-clelsb3 32973* Remove dependency on ax-ext 2631 (and df-cleq 2644) from clelsb3 2758. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
 
Theorembj-hblem 32974* Remove dependency on ax-ext 2631 (and df-cleq 2644) from hblem 2760. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝑧𝐴 → ∀𝑥 𝑧𝐴)
 
Theorembj-nfcjust 32975* Remove dependency on ax-ext 2631 (and df-cleq 2644 and ax-13 2282) from nfcjust 2781. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
 
Theorembj-nfcrii 32976* Remove dependency on ax-ext 2631 (and df-cleq 2644) from nfcrii 2786. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 
Theorembj-nfcri 32977* Remove dependency on ax-ext 2631 (and df-cleq 2644) from nfcri 2787. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝐴       𝑥 𝑦𝐴
 
Theorembj-nfnfc 32978 Remove dependency on ax-ext 2631 (and df-cleq 2644) from nfnfc 2803. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝐴       𝑥𝑦𝐴
 
Theorembj-vexwt 32979 Closed form of bj-vexw 32980. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) Use bj-vexwvt 32981 instead when sufficient. (New usage is discouraged.)
(∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
 
Theorembj-vexw 32980 If 𝜑 is a theorem, then any set belongs to the class {𝑥𝜑}. Therefore, {𝑥𝜑} is "a" universal class.

This is the closest one can get to defining a universal class, or proving vex 3234, without using ax-ext 2631. Note that this theorem has no dv condition and does not use df-clel 2647 nor df-cleq 2644 either: only first-order logic and df-clab 2638.

Without ax-ext 2631, one cannot define "the" universal class, since one could not prove for instance the justification theorem {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} (see vjust 3232). Indeed, in order to prove any equality of classes, one needs df-cleq 2644, which has ax-ext 2631 as a hypothesis. Therefore, the classes {𝑥 ∣ ⊤}, {𝑦 ∣ (𝜑𝜑)}, {𝑧 ∣ (∀𝑡𝑡 = 𝑡 → ∀𝑡𝑡 = 𝑡)} and countless others are all universal classes whose equality one cannot prove without ax-ext 2631. See also bj-issetw 32985.

A version with a dv condition between 𝑥 and 𝑦 and not requiring ax-13 2282 is proved as bj-vexwv 32982, while the degenerate instance is a simple consequence of abid 2639. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.) Use bj-vexwv 32982 instead when sufficient. (New usage is discouraged.)

𝜑       𝑦 ∈ {𝑥𝜑}
 
Theorembj-vexwvt 32981* Closed form of bj-vexwv 32982 and version of bj-vexwt 32979 with a dv condition, which does not require ax-13 2282. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
 
Theorembj-vexwv 32982* Version of bj-vexw 32980 with a dv condition, which does not require ax-13 2282. The degenerate instance of bj-vexw 32980 is a simple consequence of abid 2639 (which does not depend on ax-13 2282 either). (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
𝜑       𝑦 ∈ {𝑥𝜑}
 
Theorembj-denotes 32983* This would be the justification for the definition of the unary predicate "E!" by ( E! 𝐴 ↔ ∃𝑥𝑥 = 𝐴) which could be interpreted as "𝐴 exists" or "𝐴 denotes". It is interesting that this justification can be proved without ax-ext 2631 nor df-cleq 2644 (but of course using df-clab 2638 and df-clel 2647). Once extensionality is postulated, then isset 3238 will prove that "existing" (as a set) is equivalent to being a member of a class.

Note that there is no dv condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2282. Actually, the proof depends only on ax-1--7 and sp 2091.

The symbol "E!" was chosen to be reminiscent of the analogous predicate in (inclusive or non-inclusive) free logic, which deals with the possibility of non-existent objects. This analogy should not be taken too far, since here there are no equality axioms for classes: they are derived from ax-ext 2631 (e.g., eqid 2651). In particular, one cannot even prove 𝑥𝑥 = 𝐴𝐴 = 𝐴.

With ax-ext 2631, the present theorem is obvious from cbvexv 2311 and eqeq1 2655 (in free logic, the same proof holds since one has equality axioms for terms). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)

(∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
 
Theorembj-issetwt 32984* Closed form of bj-issetw 32985. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
(∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
 
Theorembj-issetw 32985* The closest one can get to isset 3238 without using ax-ext 2631. See also bj-vexw 32980. Note that the only dv condition is between 𝑦 and 𝐴. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
𝜑       (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴)
 
Theorembj-elissetv 32986* Version of bj-elisset 32987 with a dv condition on 𝑥, 𝑉. This proof uses only df-ex 1745, ax-gen 1762, ax-4 1777 and df-clel 2647 on top of propositional calculus. Prefer its use over bj-elisset 32987 when sufficient. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
 
Theorembj-elisset 32987* Remove from elisset 3246 dependency on ax-ext 2631 (and on df-cleq 2644 and df-v 3233). This proof uses only df-clab 2638 and df-clel 2647 on top of first-order logic. It only requires ax-1--7 and sp 2091. Use bj-elissetv 32986 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
 
Theorembj-issetiv 32988* Version of bj-isseti 32989 with a dv condition on 𝑥, 𝑉. This proof uses only df-ex 1745, ax-gen 1762, ax-4 1777 and df-clel 2647 on top of propositional calculus. Prefer its use over bj-isseti 32989 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
𝐴𝑉       𝑥 𝑥 = 𝐴
 
Theorembj-isseti 32989* Remove from isseti 3240 dependency on ax-ext 2631 (and on df-cleq 2644 and df-v 3233). This proof uses only df-clab 2638 and df-clel 2647 on top of first-order logic. It only uses ax-12 2087 among the auxiliary logical axioms. The hypothesis uses 𝑉 instead of V for extra generality. This is indeed more general as long as elex 3243 is not available. Use bj-issetiv 32988 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
𝐴𝑉       𝑥 𝑥 = 𝐴
 
Theorembj-ralvw 32990 A weak version of ralv 3250 not using ax-ext 2631 (nor df-cleq 2644, df-clel 2647, df-v 3233), but using ax-13 2282. For the sake of illustration, the next theorem bj-rexvwv 32991, a weak version of rexv 3251, has a dv condition and avoids dependency on ax-13 2282, while the analogues for reuv 3252 and rmov 3253 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝜓       (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥𝜑)
 
Theorembj-rexvwv 32991* A weak version of rexv 3251 not using ax-ext 2631 (nor df-cleq 2644, df-clel 2647, df-v 3233) with an additional dv condition to avoid dependency on ax-13 2282 as well. See bj-ralvw 32990. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝜓       (∃𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-rababwv 32992* A weak version of rabab 3254 not using df-clel 2647 nor df-v 3233 (but requiring ax-ext 2631). A version without dv condition is provable by replacing bj-vexwv 32982 with bj-vexw 32980 in the proof, hence requiring ax-13 2282. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝜓       {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥𝜑}
 
Theorembj-ralcom4 32993* Remove from ralcom4 3255 dependency on ax-ext 2631 and ax-13 2282 (and on df-or 384, df-an 385, df-tru 1526, df-sb 1938, df-clab 2638, df-cleq 2644, df-clel 2647, df-nfc 2782, df-v 3233). This proof uses only df-ral 2946 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
 
Theorembj-rexcom4 32994* Remove from rexcom4 3256 dependency on ax-ext 2631 and ax-13 2282 (and on df-or 384, df-tru 1526, df-sb 1938, df-clab 2638, df-cleq 2644, df-clel 2647, df-nfc 2782, df-v 3233). This proof uses only df-rex 2947 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
(∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
 
Theorembj-rexcom4a 32995* Remove from rexcom4a 3257 dependency on ax-ext 2631 and ax-13 2282 (and on df-or 384, df-sb 1938, df-clab 2638, df-cleq 2644, df-clel 2647, df-nfc 2782, df-v 3233). This proof uses only df-rex 2947 on top of first-order logic. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
(∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
 
Theorembj-rexcom4bv 32996* Version of bj-rexcom4b 32997 with a dv condition on 𝑥, 𝑉, hence removing dependency on df-sb 1938 and df-clab 2638 (so that it depends on df-clel 2647 and df-rex 2947 only on top of first-order logic). Prefer its use over bj-rexcom4b 32997 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
𝐵𝑉       (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
 
Theorembj-rexcom4b 32997* Remove from rexcom4b 3258 dependency on ax-ext 2631 and ax-13 2282 (and on df-or 384, df-cleq 2644, df-nfc 2782, df-v 3233). The hypothesis uses 𝑉 instead of V (see bj-isseti 32989 for the motivation). Use bj-rexcom4bv 32996 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝐵𝑉       (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
 
Theorembj-ceqsalt0 32998 The FOL content of ceqsalt 3259. Lemma for bj-ceqsalt 33000 and bj-ceqsaltv 33001. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜃 → (𝜑𝜓)) ∧ ∃𝑥𝜃) → (∀𝑥(𝜃𝜑) ↔ 𝜓))
 
Theorembj-ceqsalt1 32999 The FOL content of ceqsalt 3259. Lemma for bj-ceqsalt 33000 and bj-ceqsaltv 33001. (TODO: consider removing if it does not add anything to bj-ceqsalt0 32998.) (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
(𝜃 → ∃𝑥𝜒)       ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) ↔ 𝜓))
 
Theorembj-ceqsalt 33000* Remove from ceqsalt 3259 dependency on ax-ext 2631 (and on df-cleq 2644 and df-v 3233). Note: this is not doable with ceqsralt 3260 (or ceqsralv 3265), which uses eleq1 2718, but the same dependence removal is possible for ceqsalg 3261, ceqsal 3263, ceqsalv 3264, cgsexg 3269, cgsex2g 3270, cgsex4g 3271, ceqsex 3272, ceqsexv 3273, ceqsex2 3275, ceqsex2v 3276, ceqsex3v 3277, ceqsex4v 3278, ceqsex6v 3279, ceqsex8v 3280, gencbvex 3281 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3282, gencbval 3283, vtoclgft 3285 (it uses , whose justification nfcjust 2781 is actually provable without ax-ext 2631, as bj-nfcjust 32975 shows) and several other vtocl* theorems (see for instance bj-vtoclg1f 33036). See also bj-ceqsaltv 33001. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
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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 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