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Theorem List for Metamath Proof Explorer - 31801-31900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-nfs1v 31801* Remove dependency on ax-13 2137 from nfs1v 2329. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
𝑥[𝑦 / 𝑥]𝜑
 
Theorembj-axext3 31802* Remove dependency on ax-13 2137 from axext3 2496. (Contributed by BJ, 12-Jul-2019.) (Proof modification is discouraged.)
(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 
Theorembj-axext4 31803* Remove dependency on ax-13 2137 from axext4 2498. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
 
Theorembj-hbab1 31804* Remove dependency on ax-13 2137 from hbab1 2503. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
 
Theorembj-nfsab1 31805* Remove dependency on ax-13 2137 from nfsab1 2504. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
𝑥 𝑦 ∈ {𝑥𝜑}
 
Theorembj-abeq2 31806* Remove dependency on ax-13 2137 from abeq2 2623. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 
Theorembj-abeq1 31807* Remove dependency on ax-13 2137 from abeq1 2624. Remark: the theorems abeq2i 2626, abeq1i 2627, abeq2d 2625 do not use ax-11 1971 or ax-13 2137. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
 
Theorembj-abbi 31808 Remove dependency on ax-13 2137 from abbi 2628. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
 
Theorembj-abbi2i 31809* Remove dependency on ax-13 2137 from abbi2i 2629. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}
 
Theorembj-abbii 31810 Remove dependency on ax-13 2137 from abbii 2630. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}
 
Theorembj-abbid 31811 Remove dependency on ax-13 2137 from abbid 2631. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 
Theorembj-abbidv 31812* Remove dependency on ax-13 2137 from abbidv 2632. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 
Theorembj-abbi2dv 31813* Remove dependency on ax-13 2137 from abbi2dv 2633. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})
 
Theorembj-abbi1dv 31814* Remove dependency on ax-13 2137 from abbi1dv 2634. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} = 𝐴)
 
Theorembj-abid2 31815* Remove dependency on ax-13 2137 from abid2 2636. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
{𝑥𝑥𝐴} = 𝐴
 
Theorembj-clabel 31816* Remove dependency on ax-13 2137 from clabel 2640 (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 31817* Remove dependency on ax-13 2137 from sbab 2641 (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 31818 Remove dependency on ax-13 2137 from nfab1 2657 (note the absence of DV conditions). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥{𝑥𝜑}
 
Theorembj-vjust 31819 Remove dependency on ax-13 2137 from vjust 3078 (note the absence of DV conditions). Soundness justification theorem for df-v 3079. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
{𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
 
Theorembj-cdeqab 31820* Remove dependency on ax-13 2137 from cdeqab 3296. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})
 
Theorembj-axrep1 31821* Remove dependency on ax-13 2137 from axrep1 4598. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))
 
Theorembj-axrep2 31822* Remove dependency on ax-13 2137 from axrep2 4599. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
 
Theorembj-axrep3 31823* Remove dependency on ax-13 2137 from axrep3 4600. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
 
Theorembj-axrep4 31824* Remove dependency on ax-13 2137 from axrep4 4601. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑧𝜑       (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
 
Theorembj-axrep5 31825* Remove dependency on ax-13 2137 from axrep5 4602. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑧𝜑       (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
 
Theorembj-axsep 31826* Remove dependency on ax-13 2137 from axsep 4606. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
Theorembj-nalset 31827* Remove dependency on ax-13 2137 from nalset 4622. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
¬ ∃𝑥𝑦 𝑦𝑥
 
Theorembj-zfpow 31828* Remove dependency on ax-13 2137 from zfpow 4669. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)
 
Theorembj-el 31829* Remove dependency on ax-13 2137 from el 4672. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑦 𝑥𝑦
 
Theorembj-dtru 31830* Remove dependency on ax-13 2137 from dtru 4682. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
¬ ∀𝑥 𝑥 = 𝑦
 
Theorembj-axc16b 31831* Remove dependency on ax-13 2137 from axc16b 4683. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 
Theorembj-eunex 31832 Remove dependency on ax-13 2137 from eunex 4684. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
 
Theorembj-dtrucor 31833* Remove dependency on ax-13 2137 from dtrucor 4726. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥 = 𝑦       𝑥𝑦
 
Theorembj-dtrucor2v 31834* Version of dtrucor2 4727 with a dv condition, which does not require ax-13 2137 (nor ax-4 1713, ax-5 1793, ax-7 1885, ax-12 1983). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦𝑥𝑦)       (𝜑 ∧ ¬ 𝜑)
 
Theorembj-dvdemo1 31835* Remove dependency on ax-13 2137 from dvdemo1 4728 (this removal is noteworthy since dvdemo1 4728 and dvdemo2 4729 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑦𝑧𝑥)
 
Theorembj-dvdemo2 31836* Remove dependency on ax-13 2137 from dvdemo2 4729 (this removal is noteworthy since dvdemo1 4728 and dvdemo2 4729 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑦𝑧𝑥)
 
20.14.4.13  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 2026, hbe1 1968, hbn1 1967, modal-5 1970.

 
Theorembj-sb3b 31837 Simplified definition of substitution when variables are distinct. This is to sb3 2247 what sb4b 2250 is to sb4 2248. Actually, one may keep only bj-sb3b 31837 and sb4b 2250 in the database, renaming them sb3 and sb4. (Contributed by BJ, 6-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
 
20.14.4.14  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 31838 Biconditional version of a form of hbae 2207 with commuted quantifiers, not requiring ax-11 1971. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑧 𝑥 = 𝑦)
 
Theorembj-hbaeb 31839 Biconditional version of hbae 2207. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧𝑥 𝑥 = 𝑦)
 
Theorembj-hbnaeb 31840 Biconditional version of hbnae 2209 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
 
Theorembj-dvv 31841 A special instance of bj-hbaeb2 31838. 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)
 
20.14.4.15  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 31586), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2184 (and equsalh 2183 and equsexh 2186). Even if only one of these two theorems holds, it should be added to the database.

 
Theorembj-equsal1t 31842 Duplication of wl-equsal1t 32400, with shorter proof. Note: wl-equsalcom 32401 is also interesting. (Contributed by BJ, 6-Oct-2018.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
 
Theorembj-equsal1ti 31843 Inference associated with bj-equsal1t 31842. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)
 
Theorembj-equsal1 31844 One direction of equsal 2182. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) → 𝜓)
 
Theorembj-equsal2 31845 One direction of equsal 2182. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓))
 
Theorembj-equsal 31846 Shorter proof of equsal 2182. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2182, but "min */exc equsal" is ok. (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
20.14.4.16  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 31847 Closed form of stdpc5 2039. (Possible to place it before 19.21t 2035 and use it to prove 19.21t 2035). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
 
Theorembj-stdpc5 31848 More direct proof of stdpc5 2039. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
 
Theorem2stdpc5 31849 A double stdpc5 2039 (one direction of PM*11.3). See also 2stdpc4 2246 and 19.21vv 37498. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) → (𝜑 → ∀𝑥𝑦𝜓))
 
Theorembj-19.21t 31850 Proof of 19.21t 2035 from stdpc5t 31847. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 
Theoremexlimii 31851 Inference associated with exlimi 2043. 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 31852 Proof of ax-11 1971 similar to PM's proof of alcom 1974 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 31856. Axiom ax-11 1971 is used in the proof only through nfa2 2085. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremax6er 31853 Another form of ax6e 2141. ( Could be placed right after ax6e 2141). (Contributed by BJ, 15-Sep-2018.)
𝑥 𝑦 = 𝑥
 
Theoremexlimiieq1 31854 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 31855 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 31856* Proof of ax-11 1971 from the standard axioms of predicate calculus, similar to PM's proof of alcom 1974 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 1971 is used in the proof only through nfal 2079, nfsb 2332, sbal 2354, sb8 2316. See also ax11-pm 31852. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
20.14.4.17  Alternate definition of substitution
 
Theorembj-sbsb 31857 Biconditional showing two possible (dual) definitions of substitution df-sb 1831 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
(((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜑)))
 
Theorembj-dfsb2 31858 Alternate (dual) definition of substitution df-sb 1831 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
([𝑦 / 𝑥]𝜑 ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜑)))
 
20.14.4.18  Lemmas for substitution
 
Theorembj-sbf3 31859 Substitution has no effect on a bound variabe (existential quantifier case); see sbf2 2274. (Contributed by BJ, 2-May-2019.)
([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-sbf4 31860 Substitution has no effect on a bound variabe (non-freeness case); see sbf2 2274. (Contributed by BJ, 2-May-2019.)
([𝑦 / 𝑥]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜑)
 
Theorembj-sbnf 31861* Move non-free predicate in and out of substitution; see sbal 2354 and sbex 2355. (Contributed by BJ, 2-May-2019.)
([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)
 
20.14.4.19  Existential uniqueness
 
Theorembj-eu3f 31862* Version of eu3v 2390 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 2390. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)
𝑦𝜑       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theorembj-eumo0 31863* Existential uniqueness implies "at most one." Used to be in the main part and deprecated in favor of eumo 2391 and mo2 2371. (Contributed by NM, 8-Jul-1994.) (Revised by BJ, 8-Jun-2019.)
𝑦𝜑       (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
20.14.4.20  First-logic: miscellaneous

Miscellaneous theorems of first-order logic.

 
Theorembj-nfdiOLD 31864 Obsolete proof temporarily kept here in view of the change of df-nf 1699 to nf2 2090. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜑)       𝑥𝜑
 
Theorembj-sbieOLD 31865 Obsolete proof temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)
 
Theorembj-sbidmOLD 31866 Obsolete proof 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 31867* Obsolete proof 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))
 
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 2494, df-clab 2501, df-cleq 2507, df-clel 2510 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 2494, df-clab 2501, df-cleq 2507, df-clel 2510 }) 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 31868 and the following theorems of this section, which are special instances of df-clab 2501, dfcleq 2508 (proved from {FOL, ax-ext 2494, df-cleq 2507 }), and df-clel 2510. 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 31869, eliminable2b 31870 and eliminable3a 31872, 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 1473, 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 2501 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 2501, ax-ext 2494 and df-cleq 2507 are sufficient (over FOL) to eliminate class terms.

To prove that { df-clab 2501, df-cleq 2507, df-clel 2510 } provides a definitional extension of {FOL, ax-ext 2494 }, 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 2501, df-cleq 2507, df-clel 2510 }. 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 2494 }. 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 2501, df-cleq 2507, df-clel 2510 }. It involves a careful case study on the structure of the proof tree.

 
Theoremeliminable1 31868 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 31869* 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 31870* 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 31871* 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 31872* 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 31873* 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 31874* Every class can be written as (is equal to) a class abstraction. cvjust 2509 is a special instance of it, but the present proof does not require ax-13 2137, contrary to cvjust 2509. This theorem requires ax-ext 2494, df-clab 2501, df-cleq 2507, df-clel 2510, 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 1473, cab 2500 and statements corresponding to defined class constructors.

UPDATE: This theorem is (almost) abid2 2636 and bj-abid2 31815, though the present proof is shorter than a proof from bj-abid2 31815 and eqcomi 2523 (and is shorter than the proof of either); plus, it is of the same form as cvjust 2509 and such a basic statement deserves to be present in both forms. Note that bj-termab 31874 shortens the proof of abid2 2636, and shortens five proofs by a total of 72 bytes. Move it to Main as "abid1" proved from abbi2i 2629? Note also that this is the form in Quine, more than abid2 2636. (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 2494. One could move all theorems from cab 2500 to df-clel 2510 (except for dfcleq 2508 and cvjust 2509) 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 2507.

Note that without ax-ext 2494, the $a-statements df-clab 2501, df-cleq 2507, and df-clel 2510 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 2494, 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 1938, wel 1939, ax-8 1940, ax-9 1947).

 
Theorembj-eleq1w 31875 Weaker version of eleq1 2580 (but more general than elequ1 1945) not depending on ax-ext 2494 (nor ax-12 1983 nor df-cleq 2507). Remark: this can also be done with eleq1i 2583, eqeltri 2588, eqeltrri 2589, eleq1a 2587, eleq1d 2576, eqeltrd 2592, eqeltrrd 2593, eqneltrd 2611, eqneltrrd 2612, nelneq 2616. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
 
Theorembj-eleq2w 31876 Weaker version of eleq2 2581 (but more general than elequ2 1952) not depending on ax-ext 2494 (nor ax-12 1983 nor df-cleq 2507). (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
 
Theorembj-clelsb3 31877* Remove dependency on ax-ext 2494 (and df-cleq 2507) from clelsb3 2620. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
 
Theorembj-hblem 31878* Remove dependency on ax-ext 2494 (and df-cleq 2507) from hblem 2622. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝑧𝐴 → ∀𝑥 𝑧𝐴)
 
Theorembj-nfcjust 31879* Remove dependency on ax-ext 2494 (and df-cleq 2507 and ax-13 2137) from nfcjust 2643. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
 
Theorembj-nfcrii 31880* Remove dependency on ax-ext 2494 (and df-cleq 2507) from nfcrii 2648. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 
Theorembj-nfcri 31881* Remove dependency on ax-ext 2494 (and df-cleq 2507) from nfcri 2649. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝐴       𝑥 𝑦𝐴
 
Theorembj-nfnfc 31882 Remove dependency on ax-ext 2494 (and df-cleq 2507) from nfnfc 2664. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝐴       𝑥𝑦𝐴
 
Theorembj-vexwt 31883 Closed form of bj-vexw 31884. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) Use bj-vexwvt 31885 instead when sufficient. (New usage is discouraged.)
(∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
 
Theorembj-vexw 31884 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 3080, without using ax-ext 2494. Note that this theorem has no dv condition and does not use df-clel 2510 nor df-cleq 2507 either: only first-order logic and df-clab 2501.

Without ax-ext 2494, one cannot define "the" universal class, since one could not prove for instance the justification theorem {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} (see vjust 3078). Indeed, in order to prove any equality of classes, one needs df-cleq 2507, which has ax-ext 2494 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 2494. See also bj-issetw 31889.

A version with a dv condition between 𝑥 and 𝑦 and not requiring ax-13 2137 is proved as bj-vexwv 31886, while the degenerate instance is a simple consequence of abid 2502. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.) Use bj-vexwv 31886 instead when sufficient. (New usage is discouraged.)

𝜑       𝑦 ∈ {𝑥𝜑}
 
Theorembj-vexwvt 31885* Closed form of bj-vexwv 31886 and version of bj-vexwt 31883 with a dv condition, which does not require ax-13 2137. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
 
Theorembj-vexwv 31886* Version of bj-vexw 31884 with a dv condition, which does not require ax-13 2137. The degenerate instance of bj-vexw 31884 is a simple consequence of abid 2502 (which does not depend on ax-13 2137 either). (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
𝜑       𝑦 ∈ {𝑥𝜑}
 
Theorembj-denotes 31887* 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 2494 nor df-cleq 2507 (but of course using df-clab 2501 and df-clel 2510). Once extensionality is postulated, then isset 3084 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 2137. Actually, the proof depends only on ax-1--7 and sp 1990.

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 2494 (e.g., eqid 2514). In particular, one cannot even prove 𝑥𝑥 = 𝐴𝐴 = 𝐴.

With ax-ext 2494, the present theorem is obvious from cbvexv 2166 and eqeq1 2518 (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 31888* Closed form of bj-issetw 31889. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
(∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
 
Theorembj-issetw 31889* The closest one can get to isset 3084 without using ax-ext 2494. See also bj-vexw 31884. 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 31890* Version of bj-elisset 31891 with a dv condition on 𝑥, 𝑉. This proof uses only df-ex 1695, ax-gen 1700, ax-4 1713 and df-clel 2510 on top of propositional calculus. Prefer its use over bj-elisset 31891 when sufficient. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
 
Theorembj-elisset 31891* Remove from elisset 3092 dependency on ax-ext 2494 (and on df-cleq 2507 and df-v 3079). This proof uses only df-clab 2501 and df-clel 2510 on top of first-order logic. It only requires ax-1--7 and sp 1990. Use bj-elissetv 31890 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 31892* Version of bj-isseti 31893 with a dv condition on 𝑥, 𝑉. This proof uses only df-ex 1695, ax-gen 1700, ax-4 1713 and df-clel 2510 on top of propositional calculus. Prefer its use over bj-isseti 31893 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 31893* Remove from isseti 3086 dependency on ax-ext 2494 (and on df-cleq 2507 and df-v 3079). This proof uses only df-clab 2501 and df-clel 2510 on top of first-order logic. It only uses ax-12 1983 among the auxiliary logical axioms. The hypothesis uses 𝑉 instead of V for extra generality. This is indeed more general as long as elex 3089 is not available. Use bj-issetiv 31892 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 31894 A weak version of ralv 3096 not using ax-ext 2494 (nor df-cleq 2507, df-clel 2510, df-v 3079), but using ax-13 2137. For the sake of illustration, the next theorem bj-rexvwv 31895, a weak version of rexv 3097, has a dv condition and avoids dependency on ax-13 2137, while the analogues for reuv 3098 and rmov 3099 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝜓       (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥𝜑)
 
Theorembj-rexvwv 31895* A weak version of rexv 3097 not using ax-ext 2494 (nor df-cleq 2507, df-clel 2510, df-v 3079) with an additional dv condition to avoid dependency on ax-13 2137 as well. See bj-ralvw 31894. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝜓       (∃𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-rababwv 31896* A weak version of rabab 3100 not using df-clel 2510 nor df-v 3079 (but requiring ax-ext 2494). A version without dv condition is provable by replacing bj-vexwv 31886 with bj-vexw 31884 in the proof, hence requiring ax-13 2137. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝜓       {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥𝜑}
 
Theorembj-ralcom4 31897* Remove from ralcom4 3101 dependency on ax-ext 2494 and ax-13 2137 (and on df-or 383, df-an 384, df-tru 1477, df-sb 1831, df-clab 2501, df-cleq 2507, df-clel 2510, df-nfc 2644, df-v 3079). This proof uses only df-ral 2805 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
 
Theorembj-rexcom4 31898* Remove from rexcom4 3102 dependency on ax-ext 2494 and ax-13 2137 (and on df-or 383, df-tru 1477, df-sb 1831, df-clab 2501, df-cleq 2507, df-clel 2510, df-nfc 2644, df-v 3079). This proof uses only df-rex 2806 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
(∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
 
Theorembj-rexcom4a 31899* Remove from rexcom4a 3103 dependency on ax-ext 2494 and ax-13 2137 (and on df-or 383, df-sb 1831, df-clab 2501, df-cleq 2507, df-clel 2510, df-nfc 2644, df-v 3079). This proof uses only df-rex 2806 on top of first-order logic. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
(∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
 
Theorembj-rexcom4bv 31900* Version of bj-rexcom4b 31901 with a dv condition on 𝑥, 𝑉, hence removing dependency on df-sb 1831 and df-clab 2501 (so that it depends on df-clel 2510 and df-rex 2806 only on top of first-order logic). Prefer its use over bj-rexcom4b 31901 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)
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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 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