P. 361 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36001-36100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-hbnaeb 36001	Biconditional version of hbnae 2429 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	bj-dvv 36002	A special instance of bj-hbaeb2 35999. 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 ↔ ∀𝑥∀𝑦 𝑥 = 𝑦)
21.17.4.14  Around ~ equsal As a rule of thumb, if a theorem of the form ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (𝜒 ↔ 𝜃) is in the database, and the "more precise" theorems ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜒 → 𝜃) and ⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜃 → 𝜒) also hold (see bj-bisym 35771), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2415 (and equsalh 2417 and equsexh 2418). Even if only one of these two theorems holds, it should be added to the database.
Theorem	bj-equsal1t 36003	Duplication of wl-equsal1t 36713, with shorter proof. If one imposes a disjoint variable condition on x,y , then one can use alequexv 2002 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 36714 is also interesting. (Contributed by BJ, 6-Oct-2018.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	bj-equsal1ti 36004	Inference associated with bj-equsal1t 36003. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)
Theorem	bj-equsal1 36005	One direction of equsal 2414. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓)
Theorem	bj-equsal2 36006	One direction of equsal 2414. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓))
Theorem	bj-equsal 36007	Shorter proof of equsal 2414. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2414, but "min */exc equsal" is ok. (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
21.17.4.15  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".
Theorem	stdpc5t 36008	Closed form of stdpc5 2199. (Possible to place it before 19.21t 2197 and use it to prove 19.21t 2197). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-stdpc5 36009	More direct proof of stdpc5 2199. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	2stdpc5 36010	A double stdpc5 2199 (one direction of PM*11.3). See also 2stdpc4 2071 and 19.21vv 43437. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-19.21t0 36011	Proof of 19.21t 2197 from stdpc5t 36008. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	exlimii 36012	Inference associated with exlimi 2208. Inferring a theorem when it is implied by an antecedent which may be true. (Contributed by BJ, 15-Sep-2018.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜑 → 𝜓)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ 𝜓
Theorem	ax11-pm 36013	Proof of ax-11 2152 similar to PM's proof of alcom 2154 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 36017. Axiom ax-11 2152 is used in the proof only through nfa2 2168. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	ax6er 36014	Commuted form of ax6e 2380. (Could be placed right after ax6e 2380). (Contributed by BJ, 15-Sep-2018.)
⊢ ∃𝑥 𝑦 = 𝑥
Theorem	exlimiieq1 36015	Inferring a theorem when it is implied by an equality which may be true. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	exlimiieq2 36016	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.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	ax11-pm2 36017*	Proof of ax-11 2152 from the standard axioms of predicate calculus, similar to PM's proof of alcom 2154 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 2152 is used in the proof only through nfal 2314, nfsb 2520, sbal 2157, sb8 2514. See also ax11-pm 36013. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
21.17.4.16  Alternate definition of substitution
Theorem	bj-sbsb 36018	Biconditional showing two possible (dual) definitions of substitution df-sb 2066 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∨ (𝑥 = 𝑦 ∧ 𝜑)))
Theorem	bj-dfsb2 36019	Alternate (dual) definition of substitution df-sb 2066 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∨ (𝑥 = 𝑦 ∧ 𝜑)))
21.17.4.17  Lemmas for substitution
Theorem	bj-sbf3 36020	Substitution has no effect on a bound variable (existential quantifier case); see sbf2 2261. (Contributed by BJ, 2-May-2019.)
⊢ ([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-sbf4 36021	Substitution has no effect on a bound variable (nonfreeness case); see sbf2 2261. (Contributed by BJ, 2-May-2019.)
⊢ ([𝑦 / 𝑥]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜑)
Theorem	bj-sbnf 36022*	Move nonfree predicate in and out of substitution; see sbal 2157 and sbex 2275. (Contributed by BJ, 2-May-2019.)
⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)
21.17.4.18  Existential uniqueness
Theorem	bj-eu3f 36023*	Version of eu3v 2562 where the disjoint variable condition is replaced with a nonfreeness 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 2562. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
21.17.4.19  First-order logic: miscellaneous Miscellaneous theorems of first-order logic.
Theorem	bj-sblem1 36024*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜒)))
Theorem	bj-sblem2 36025*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → ((∃𝑥𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-sblem 36026*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ (∀𝑥(𝜑 → (𝜓 ↔ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜒)))
Theorem	bj-sbievw1 36027*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → 𝜓))
Theorem	bj-sbievw2 36028*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ ([𝑦 / 𝑥](𝜓 → 𝜑) → (𝜓 → [𝑦 / 𝑥]𝜑))
Theorem	bj-sbievw 36029*	Lemma for substitution. Closed form of equsalvw 2005 and sbievw 2093. (Contributed by BJ, 23-Jul-2023.)
⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
Theorem	bj-sbievv 36030	Version of sbie 2499 with a second nonfreeness hypothesis and shorter proof. (Contributed by BJ, 18-Jul-2023.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
Theorem	bj-moeub 36031	Uniqueness is equivalent to existence being equivalent to unique existence. (Contributed by BJ, 14-Oct-2022.)
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
Theorem	bj-sbidmOLD 36032	Obsolete proof of sbidm 2507 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)
Theorem	bj-dvelimdv 36033*	Deduction form of dvelim 2448 with disjoint variable conditions. Uncurried (imported) form of bj-dvelimdv1 36034. 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 weaken the implicit substitution hypothesis by adding the antecedent 𝜑 but this typically does not make the theorem much more useful. Similarly, one could use nonfreeness hypotheses instead of disjoint variable 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 (𝑥, 𝑧) since in the proof nfv 1915 can be replaced with nfal 2314 followed by nfn 1858. Remark: nfald 2319 uses ax-11 2152; it might be possible to inline and use ax11w 2124 instead, but there is still a use via 19.12 2318 anyway. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓))    ⇒   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
Theorem	bj-dvelimdv1 36034*	Curried (exported) form of bj-dvelimdv 36033 (of course, one is directly provable from the other, but we keep this proof for illustration purposes). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓))
Theorem	bj-dvelimv 36035*	A version of dvelim 2448 using the "nonfree" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜑))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
Theorem	bj-nfeel2 36036*	Nonfreeness in a membership statement. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧)
Theorem	bj-axc14nf 36037	Proof of a version of axc14 2460 using the "nonfree" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦))
Theorem	bj-axc14 36038	Alternate proof of axc14 2460 (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 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
Theorem	mobidvALT 36039*	Alternate proof of mobidv 2541 directly from its analogues albidv 1921 and exbidv 1922, using deduction style. Note the proof structure, similar to mobi 2539. (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. Remove dependency on ax-6 1969, ax-7 2009, ax-12 2169 by adapting proof of mobid 2542. (Revised by BJ, 26-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Theorem	sbn1ALT 36040	Alternate proof of sbn1 2103, not using the false constant. (Contributed by BJ, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)
21.17.5  Set theory
21.17.5.1  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 2701, df-clab 2708, df-cleq 2722, df-clel 2808 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 in set.mm. It states: every formula in the language of FOL + ∈ + class terms, but without class variables, is provably equivalent (over {FOL, ax-ext 2701, df-clab 2708, df-cleq 2722, df-clel 2808 }) 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 six following forms: for equality, 𝑥 = {𝑦 ∣ 𝜑}, {𝑥 ∣ 𝜑} = 𝑦, {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}, and for membership, 𝑦 ∈ {𝑥 ∣ 𝜑}, {𝑥 ∣ 𝜑} ∈ 𝑦, {𝑥 ∣ 𝜑} ∈ {𝑦 ∣ 𝜓}. These cases are dealt with by eliminable-veqab 36048, eliminable-abeqv 36049, eliminable-abeqab 36050, eliminable-velab 36047, eliminable-abelv 36051, eliminable-abelab 36052 respectively, which are all proved from {FOL, ax-ext 2701, df-clab 2708, df-cleq 2722, df-clel 2808 }. (Details on the proof of the above six theorems. To understand how they were systematically proved, look at the theorems "eliminablei" below, which are special instances of df-clab 2708, dfcleq 2723 (proved from {FOL, ax-ext 2701, df-cleq 2722 }), and dfclel 2809 (proved from {FOL, df-clel 2808 }). 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 36042, eliminable2b 36043 and eliminable3a 36045, 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 1538, 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 ∈ {𝑥 ∣ 𝜑}, then df-clab 2708 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 ∈ {𝑥 ∣ 𝜑} and equalities, then df-clab 2708, ax-ext 2701 and df-cleq 2722 are sufficient (over FOL) to eliminate class terms. To prove that { df-clab 2708, df-cleq 2722, df-clel 2808 } provides a definitional extension of {FOL, ax-ext 2701 }, one needs to prove both 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 2708, df-cleq 2722, df-clel 2808 }. 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 2701 }. 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 2708, df-cleq 2722, df-clel 2808 }. It involves a careful case study on the structure of the proof tree.
Theorem	eliminable1 36041	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 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
Theorem	eliminable2a 36042*	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 = {𝑦 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑}))
Theorem	eliminable2b 36043*	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 ∣ 𝜑} = 𝑦 ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ 𝑦))
Theorem	eliminable2c 36044*	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 ∣ 𝜑} = {𝑦 ∣ 𝜓} ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓}))
Theorem	eliminable3a 36045*	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 ∣ 𝜑} ∈ 𝑦 ↔ ∃𝑧(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 ∈ 𝑦))
Theorem	eliminable3b 36046*	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 ∣ 𝜑} ∈ {𝑦 ∣ 𝜓} ↔ ∃𝑧(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 ∈ {𝑦 ∣ 𝜓}))
Theorem	eliminable-velab 36047	A theorem used to prove the base case of the Eliminability Theorem (see section comment): variable belongs to abstraction. (Contributed by BJ, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
Theorem	eliminable-veqab 36048*	A theorem used to prove the base case of the Eliminability Theorem (see section comment): variable equals abstraction. (Contributed by BJ, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = {𝑦 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ [𝑧 / 𝑦]𝜑))
Theorem	eliminable-abeqv 36049*	A theorem used to prove the base case of the Eliminability Theorem (see section comment): abstraction equals variable. (Contributed by BJ, 30-Apr-2024.) Beware not to use symmetry of class equality. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({𝑥 ∣ 𝜑} = 𝑦 ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ 𝑦))
Theorem	eliminable-abeqab 36050*	A theorem used to prove the base case of the Eliminability Theorem (see section comment): abstraction equals abstraction. (Contributed by BJ, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓))
Theorem	eliminable-abelv 36051*	A theorem used to prove the base case of the Eliminability Theorem (see section comment): abstraction belongs to variable. (Contributed by BJ, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({𝑥 ∣ 𝜑} ∈ 𝑦 ↔ ∃𝑧(∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ 𝑧 ∈ 𝑦))
Theorem	eliminable-abelab 36052*	A theorem used to prove the base case of the Eliminability Theorem (see section comment): abstraction belongs to abstraction. (Contributed by BJ, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({𝑥 ∣ 𝜑} ∈ {𝑦 ∣ 𝜓} ↔ ∃𝑧(∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ [𝑧 / 𝑦]𝜓))
21.17.5.2  Classes without the axiom of extensionality A few results about classes can be proved without using ax-ext 2701. One could move all theorems from cab 2707 to df-clel 2808 (except for dfcleq 2723 and cvjust 2724) 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 2722. Note that without ax-ext 2701, the $a-statements df-clab 2708, df-cleq 2722, and df-clel 2808 are no longer eliminable (see previous section) (but PROBABLY df-clab 2708 is still conservative , while df-cleq 2722 and df-clel 2808 are not). This is not a reason not to study what is provable with them but without ax-ext 2701, 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 2104, wel 2105, ax-8 2106, ax-9 2114). Remark: the weakening of eleq1 2819 / eleq2 2820 to eleq1w 2814 / eleq2w 2815 can also be done with eleq1i 2822, eqeltri 2827, eqeltrri 2828, eleq1a 2826, eleq1d 2816, eqeltrd 2831, eqeltrrd 2832, eqneltrd 2851, eqneltrrd 2852, nelneq 2855. Remark: possibility to remove dependency on ax-10 2135, ax-11 2152, ax-13 2369 from nfcri 2888 and theorems using it if one adds a disjoint variable condition (that theorem is typically used with dummy variables, so the disjoint variable condition addition is not very restrictive), and then shorten nfnfc 2913.
Theorem	bj-denoteslem 36053*	Lemma for bj-denotes 36054. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)
⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
Theorem	bj-denotes 36054*	This would be the justification theorem for the definition of the unary predicate "E!" by ⊢ ( E! 𝐴 ↔ ∃𝑥𝑥 = 𝐴) which could be interpreted as "𝐴 exists" (as a set) or "𝐴 denotes" (in the sense of free logic). A shorter proof using bitri 274 (to add an intermediate proposition ∃𝑧𝑧 = 𝐴 with a fresh 𝑧), cbvexvw 2038, and eqeq1 2734, requires the core axioms and { ax-9 2114, ax-ext 2701, df-cleq 2722 } whereas this proof requires the core axioms and { ax-8 2106, df-clab 2708, df-clel 2808 }. Theorem bj-issetwt 36057 proves that "existing" is equivalent to being a member of a class abstraction. It also requires, with the present proof, { ax-8 2106, df-clab 2708, df-clel 2808 } (whereas with the shorter proof from cbvexvw 2038 and eqeq1 2734 it would require { ax-8 2106, ax-9 2114, ax-ext 2701, df-clab 2708, df-cleq 2722, df-clel 2808 }). That every class is equal to a class abstraction is proved by abid1 2868, which requires { ax-8 2106, ax-9 2114, ax-ext 2701, df-clab 2708, df-cleq 2722, df-clel 2808 }. Note that there is no disjoint variable condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2369. Actually, the proof depends only on the logical axioms ax-1 6 through ax-7 2009 and sp 2174. 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 nonexistent objects. This analogy should not be taken too far, since here there are no equality axioms for classes: these are derived from ax-ext 2701 and df-cleq 2722 (e.g., eqid 2730 and eqeq1 2734). In particular, one cannot even prove ⊢ ∃𝑥𝑥 = 𝐴 ⇒ ⊢ 𝐴 = 𝐴 without ax-ext 2701 and df-cleq 2722. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
Theorem	bj-issettru 36055*	Weak version of isset 3485 without ax-ext 2701. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)
⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
Theorem	bj-elabtru 36056	This is as close as we can get to proving extensionality for "the" "universal" class without ax-ext 2701. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ {𝑥 ∣ ⊤} ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
Theorem	bj-issetwt 36057*	Closed form of bj-issetw 36058. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
Theorem	bj-issetw 36058*	The closest one can get to isset 3485 without using ax-ext 2701. See also vexw 2713. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3485 using eleq2i 2823 (which requires ax-ext 2701 and df-cleq 2722). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)
Theorem	bj-elissetALT 36059*	Alternate proof of elisset 2813. This is essentially the same proof as seen by inlining bj-denotes 36054 and bj-denoteslem 36053. Use elissetv 2812 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
Theorem	bj-issetiv 36060*	Version of bj-isseti 36061 with a disjoint variable condition on 𝑥, 𝑉. The hypothesis uses 𝑉 instead of V for extra generality. This is indeed more general than isseti 3488 as long as elex 3491 is not available (and the non-dependence of bj-issetiv 36060 on special properties of the universal class V is obvious). Prefer its use over bj-isseti 36061 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∃𝑥 𝑥 = 𝐴
Theorem	bj-isseti 36061*	Version of isseti 3488 with a class variable 𝑉 in the hypothesis instead of V for extra generality. This is indeed more general than isseti 3488 as long as elex 3491 is not available (and the non-dependence of bj-isseti 36061 on special properties of the universal class V is obvious). Use bj-issetiv 36060 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∃𝑥 𝑥 = 𝐴
Theorem	bj-ralvw 36062	A weak version of ralv 3497 not using ax-ext 2701 (nor df-cleq 2722, df-clel 2808, df-v 3474), and only core FOL axioms. See also bj-rexvw 36063. The analogues for reuv 3499 and rmov 3500 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ 𝜓    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥𝜑)
Theorem	bj-rexvw 36063	A weak version of rexv 3498 not using ax-ext 2701 (nor df-cleq 2722, df-clel 2808, df-v 3474), and only core FOL axioms. See also bj-ralvw 36062. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ 𝜓    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-rababw 36064	A weak version of rabab 3501 not using df-clel 2808 nor df-v 3474 (but requiring ax-ext 2701) nor ax-12 2169. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ 𝜓    ⇒   ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑}
Theorem	bj-rexcom4bv 36065*	Version of rexcom4b 3502 and bj-rexcom4b 36066 with a disjoint variable condition on 𝑥, 𝑉, hence removing dependency on df-sb 2066 and df-clab 2708 (so that it depends on df-clel 2808 and df-rex 3069 only on top of first-order logic). Prefer its use over bj-rexcom4b 36066 when sufficient (in particular when 𝑉 is substituted for V). Note the 𝑉 in the hypothesis instead of V. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
⊢ 𝐵 ∈ 𝑉    ⇒   ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)
Theorem	bj-rexcom4b 36066*	Remove from rexcom4b 3502 dependency on ax-ext 2701 and ax-13 2369 (and on df-or 844, df-cleq 2722, df-nfc 2883, df-v 3474). The hypothesis uses 𝑉 instead of V (see bj-isseti 36061 for the motivation). Use bj-rexcom4bv 36065 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ 𝐵 ∈ 𝑉    ⇒   ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)
Theorem	bj-ceqsalt0 36067	The FOL content of ceqsalt 3504. Lemma for bj-ceqsalt 36069 and bj-ceqsaltv 36070. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜃 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥𝜃) → (∀𝑥(𝜃 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalt1 36068	The FOL content of ceqsalt 3504. Lemma for bj-ceqsalt 36069 and bj-ceqsaltv 36070. TODO: consider removing if it does not add anything to bj-ceqsalt0 36067. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
⊢ (𝜃 → ∃𝑥𝜒)    ⇒   ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalt 36069*	Remove from ceqsalt 3504 dependency on ax-ext 2701 (and on df-cleq 2722 and df-v 3474). Note: this is not doable with ceqsralt 3505 (or ceqsralv 3512), which uses eleq1 2819, but the same dependence removal is possible for ceqsalg 3506, ceqsal 3508, ceqsalv 3510, cgsexg 3517, cgsex2g 3518, cgsex4g 3519, ceqsex 3522, ceqsexv 3524, ceqsex2 3528, ceqsex2v 3529, ceqsex3v 3530, ceqsex4v 3531, ceqsex6v 3532, ceqsex8v 3533, gencbvex 3534 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3535, gencbval 3536, vtoclgft 3539 (it uses Ⅎ, whose justification nfcjust 2882 does not use ax-ext 2701) and several other vtocl* theorems (see for instance bj-vtoclg1f 36101). See also bj-ceqsaltv 36070. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsaltv 36070*	Version of bj-ceqsalt 36069 with a disjoint variable condition on 𝑥, 𝑉, removing dependency on df-sb 2066 and df-clab 2708. Prefer its use over bj-ceqsalt 36069 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalg0 36071	The FOL content of ceqsalg 3506. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜒 → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalg 36072*	Remove from ceqsalg 3506 dependency on ax-ext 2701 (and on df-cleq 2722 and df-v 3474). See also bj-ceqsalgv 36074. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalgALT 36073*	Alternate proof of bj-ceqsalg 36072. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalgv 36074*	Version of bj-ceqsalg 36072 with a disjoint variable condition on 𝑥, 𝑉, removing dependency on df-sb 2066 and df-clab 2708. Prefer its use over bj-ceqsalg 36072 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalgvALT 36075*	Alternate proof of bj-ceqsalgv 36074. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsal 36076*	Remove from ceqsal 3508 dependency on ax-ext 2701 (and on df-cleq 2722, df-v 3474, df-clab 2708, df-sb 2066). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	bj-ceqsalv 36077*	Remove from ceqsalv 3510 dependency on ax-ext 2701 (and on df-cleq 2722, df-v 3474, df-clab 2708, df-sb 2066). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	bj-spcimdv 36078*	Remove from spcimdv 3582 dependency on ax-9 2114, ax-10 2135, ax-11 2152, ax-13 2369, ax-ext 2701, df-cleq 2722 (and df-nfc 2883, df-v 3474, df-or 844, df-tru 1542, df-nf 1784). For an even more economical version, see bj-spcimdvv 36079. (Contributed by BJ, 30-Nov-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-spcimdvv 36079*	Remove from spcimdv 3582 dependency on ax-7 2009, ax-8 2106, ax-10 2135, ax-11 2152, ax-12 2169 ax-13 2369, ax-ext 2701, df-cleq 2722, df-clab 2708 (and df-nfc 2883, df-v 3474, df-or 844, df-tru 1542, df-nf 1784) at the price of adding a disjoint variable condition on 𝑥, 𝐵 (but in usages, 𝑥 is typically a dummy, hence fresh, variable). For the version without this disjoint variable condition, see bj-spcimdv 36078. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
21.17.5.3  Characterization among sets versus among classes
Theorem	elelb 36080	Equivalence between two common ways to characterize elements of a class 𝐵: the LHS says that sets are elements of 𝐵 if and only if they satisfy 𝜑 while the RHS says that classes are elements of 𝐵 if and only if they are sets and satisfy 𝜑. Therefore, the LHS is a characterization among sets while the RHS is a characterization among classes. Note that the LHS is often formulated using a class variable instead of the universe V while this is not possible for the RHS (apart from using 𝐵 itself, which would not be very useful). (Contributed by BJ, 26-Feb-2023.)
⊢ ((𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜑)) ↔ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜑)))
Theorem	bj-pwvrelb 36081	Characterization of the elements of the powerclass of the cartesian square of the universal class: they are exactly the sets which are binary relations. (Contributed by BJ, 16-Dec-2023.)
⊢ (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴))
21.17.5.4  The nonfreeness quantifier for classes In this section, we prove the symmetry of the nonfreeness quantifier for classes.
Theorem	bj-nfcsym 36082	The nonfreeness quantifier for classes defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 5372 with additional axioms; see also nfcv 2901). This could be proved from aecom 2424 and nfcvb 5373 but the latter requires a domain with at least two objects (hence uses extra axioms). (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid use of eqcomd 2736 instead of equcomd 2020; removing dependency on ax-ext 2701 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2920, eleq2d 2817 (using elequ2 2119), nfcvf 2930, dvelimc 2929, dvelimdc 2928, nfcvf2 2931. (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)
21.17.5.5  Lemmas for class substitution Some useful theorems for dealing with substitutions: sbbi 2302, sbcbig 3830, sbcel1g 4412, sbcel2 4414, sbcel12 4407, sbceqg 4408, csbvarg 4430.
Theorem	bj-sbeqALT 36083*	Substitution in an equality (use the more general version bj-sbeq 36084 instead, without disjoint variable condition). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)
Theorem	bj-sbeq 36084	Distribute proper substitution through an equality relation. (See sbceqg 4408). (Contributed by BJ, 6-Oct-2018.)
⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)
Theorem	bj-sbceqgALT 36085	Distribute proper substitution through an equality relation. Alternate proof of sbceqg 4408. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 4408, but the Metamath program "MM-PA> MINIMIZE_WITH * / EXCEPT sbceqg" command is ok. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
Theorem	bj-csbsnlem 36086*	Lemma for bj-csbsn 36087 (in this lemma, 𝑥 cannot occur in 𝐴). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}
Theorem	bj-csbsn 36087	Substitution in a singleton. (Contributed by BJ, 6-Oct-2018.)
⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}
Theorem	bj-sbel1 36088*	Version of sbcel1g 4412 when substituting a set. (Note: one could have a corresponding version of sbcel12 4407 when substituting a set, but the point here is that the antecedent of sbcel1g 4412 is not needed when substituting a set.) (Contributed by BJ, 6-Oct-2018.)
⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵)
Theorem	bj-abv 36089	The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)
Theorem	bj-abvALT 36090	Alternate version of bj-abv 36089; shorter but uses ax-8 2106. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)
Theorem	bj-ab0 36091	The class of sets verifying a falsity is the empty set (closed form of abf 4401). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
Theorem	bj-abf 36092	Shorter proof of abf 4401 (which should be kept as abfALT). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ {𝑥 ∣ 𝜑} = ∅
Theorem	bj-csbprc 36093	More direct proof of csbprc 4405 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
21.17.5.6  Removing some axiom requirements and disjoint variable conditions
Theorem	bj-exlimvmpi 36094*	A Fol lemma (exlimiv 1931 followed by mpi 20). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpi 36095	Lemma for bj-vtoclg1f1 36100 (an instance of this lemma is a version of bj-vtoclg1f1 36100 where 𝑥 and 𝑦 are identified). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpbi 36096	Lemma for theorems of the vtoclg 3541 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpbir 36097	Lemma for theorems of the vtoclg 3541 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ (∃𝑥𝜒 → 𝜑)
Theorem	bj-vtoclf 36098*	Remove dependency on ax-ext 2701, df-clab 2708 and df-cleq 2722 (and df-sb 2066 and df-v 3474) from vtoclf 3550. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ 𝑉    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	bj-vtocl 36099*	Remove dependency on ax-ext 2701, df-clab 2708 and df-cleq 2722 (and df-sb 2066 and df-v 3474) from vtocl 3544. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝑉    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	bj-vtoclg1f1 36100*	The FOL content of vtoclg1f 3557 (hence not using ax-ext 2701, df-cleq 2722, df-nfc 2883, df-v 3474). Note the weakened "major" hypothesis and the disjoint variable condition between 𝑥 and 𝐴 (needed since the nonfreeness quantifier for classes is not available without ax-ext 2701; as a byproduct, this dispenses with ax-11 2152 and ax-13 2369). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑦 𝑦 = 𝐴 → 𝜓)