P. 369 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36801-36900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-hbaeb 36801	Biconditional version of hbae 2433. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)
Theorem	bj-hbnaeb 36802	Biconditional version of hbnae 2434 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	bj-dvv 36803	A special instance of bj-hbaeb2 36800. 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)
21.19.4.14  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 36572), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2420 (and equsalh 2422 and equsexh 2423). Even if only one of these two theorems holds, it should be added to the database.
Theorem	bj-equsal1t 36804	Duplication of wl-equsal1t 37522, with shorter proof. If one imposes a disjoint variable condition on x,y , then one can use alequexv 1997 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 37523 is also interesting. (Contributed by BJ, 6-Oct-2018.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	bj-equsal1ti 36805	Inference associated with bj-equsal1t 36804. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)
Theorem	bj-equsal1 36806	One direction of equsal 2419. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓)
Theorem	bj-equsal2 36807	One direction of equsal 2419. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓))
Theorem	bj-equsal 36808	Shorter proof of equsal 2419. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2419, but "min */exc equsal" is ok. (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
21.19.4.15  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 36809	Closed form of stdpc5 2205. (Possible to place it before 19.21t 2203 and use it to prove 19.21t 2203). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-stdpc5 36810	More direct proof of stdpc5 2205. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	2stdpc5 36811	A double stdpc5 2205 (one direction of PM*11.3). See also 2stdpc4 2067 and 19.21vv 44371. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-19.21t0 36812	Proof of 19.21t 2203 from stdpc5t 36809. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	exlimii 36813	Inference associated with exlimi 2214. 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 36814	Proof of ax-11 2154 similar to PM's proof of alcom 2156 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 36818. Axiom ax-11 2154 is used in the proof only through nfa2 2173. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	ax6er 36815	Commuted form of ax6e 2385. (Could be placed right after ax6e 2385). (Contributed by BJ, 15-Sep-2018.)
⊢ ∃𝑥 𝑦 = 𝑥
Theorem	exlimiieq1 36816	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 36817	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 36818*	Proof of ax-11 2154 from the standard axioms of predicate calculus, similar to PM's proof of alcom 2156 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 2154 is used in the proof only through nfal 2321, nfsb 2525, sbal 2166, sb8 2519. See also ax11-pm 36814. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
21.19.4.16  Alternate definition of substitution
Theorem	bj-sbsb 36819	Biconditional showing two possible (dual) definitions of substitution df-sb 2062 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 36820	Alternate (dual) definition of substitution df-sb 2062 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∨ (𝑥 = 𝑦 ∧ 𝜑)))
21.19.4.17  Lemmas for substitution
Theorem	bj-sbf3 36821	Substitution has no effect on a bound variable (existential quantifier case); see sbf2 2269. (Contributed by BJ, 2-May-2019.)
⊢ ([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-sbf4 36822	Substitution has no effect on a bound variable (nonfreeness case); see sbf2 2269. (Contributed by BJ, 2-May-2019.)
⊢ ([𝑦 / 𝑥]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜑)
21.19.4.18  Existential uniqueness
Theorem	bj-eu3f 36823*	Version of eu3v 2567 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 2567. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
21.19.4.19  First-order logic: miscellaneous Miscellaneous theorems of first-order logic.
Theorem	bj-sblem1 36824*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜒)))
Theorem	bj-sblem2 36825*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → ((∃𝑥𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-sblem 36826*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ (∀𝑥(𝜑 → (𝜓 ↔ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜒)))
Theorem	bj-sbievw1 36827*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → 𝜓))
Theorem	bj-sbievw2 36828*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ ([𝑦 / 𝑥](𝜓 → 𝜑) → (𝜓 → [𝑦 / 𝑥]𝜑))
Theorem	bj-sbievw 36829*	Lemma for substitution. Closed form of equsalvw 2000 and sbievw 2090. (Contributed by BJ, 23-Jul-2023.)
⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
Theorem	bj-sbievv 36830	Version of sbie 2504 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 36831	Uniqueness is equivalent to existence being equivalent to unique existence. (Contributed by BJ, 14-Oct-2022.)
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
Theorem	bj-sbidmOLD 36832	Obsolete proof of sbidm 2512 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 36833*	Deduction form of dvelim 2453 with disjoint variable conditions. Uncurried (imported) form of bj-dvelimdv1 36834. 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 1911 can be replaced with nfal 2321 followed by nfn 1854. Remark: nfald 2326 uses ax-11 2154; it might be possible to inline and use ax11w 2127 instead, but there is still a use via 19.12 2325 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 36834*	Curried (exported) form of bj-dvelimdv 36833 (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 36835*	A version of dvelim 2453 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 36836*	Nonfreeness in a membership statement. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧)
Theorem	bj-axc14nf 36837	Proof of a version of axc14 2465 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 36838	Alternate proof of axc14 2465 (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 36839*	Alternate proof of mobidv 2546 directly from its analogues albidv 1917 and exbidv 1918, using deduction style. Note the proof structure, similar to mobi 2544. (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. Remove dependency on ax-6 1964, ax-7 2004, ax-12 2174 by adapting proof of mobid 2547. (Revised by BJ, 26-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Theorem	sbn1ALT 36840	Alternate proof of sbn1 2104, 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)
21.19.5  Set theory
21.19.5.1  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 2705, df-clab 2712, df-cleq 2726, df-clel 2813 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 2705, df-clab 2712, df-cleq 2726, df-clel 2813 }) 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 36848, eliminable-abeqv 36849, eliminable-abeqab 36850, eliminable-velab 36847, eliminable-abelv 36851, eliminable-abelab 36852 respectively, which are all proved from {FOL, ax-ext 2705, df-clab 2712, df-cleq 2726, df-clel 2813 }. (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 2712, dfcleq 2727 (proved from {FOL, ax-ext 2705, df-cleq 2726 }), and dfclel 2814 (proved from {FOL, df-clel 2813 }). 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 36842, eliminable2b 36843 and eliminable3a 36845, 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 1535, 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 2712 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 2712, ax-ext 2705 and df-cleq 2726 are sufficient (over FOL) to eliminate class terms. To prove that { df-clab 2712, df-cleq 2726, df-clel 2813 } provides a definitional extension of {FOL, ax-ext 2705 }, 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 2712, df-cleq 2726, df-clel 2813 }. 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 2705 }. 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 2712, df-cleq 2726, df-clel 2813 }. It involves a careful case study on the structure of the proof tree.
Theorem	eliminable1 36841	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 36842*	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 36843*	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 36844*	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 36845*	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 36846*	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 36847	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 36848*	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 36849*	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 36850*	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 36851*	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 36852*	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))
21.19.5.2  Classes without the axiom of extensionality A few results about classes can be proved without using ax-ext 2705. One could move all theorems from cab 2711 to df-clel 2813 (except for dfcleq 2727 and cvjust 2728) 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 2726. Note that without ax-ext 2705, the $a-statements df-clab 2712, df-cleq 2726, and df-clel 2813 are no longer eliminable (see previous section) (but PROBABLY df-clab 2712 is still conservative , while df-cleq 2726 and df-clel 2813 are not). This is not a reason not to study what is provable with them but without ax-ext 2705, 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 2105, wel 2106, ax-8 2107, ax-9 2115). Remark: the weakening of eleq1 2826 / eleq2 2827 to eleq1w 2821 / eleq2w 2822 can also be done with eleq1i 2829, eqeltri 2834, eqeltrri 2835, eleq1a 2833, eleq1d 2823, eqeltrd 2838, eqeltrrd 2839, eqneltrd 2858, eqneltrrd 2859, nelneq 2862. Remark: possibility to remove dependency on ax-10 2138, ax-11 2154, ax-13 2374 from nfcri 2894 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 2915.
Theorem	bj-denoteslem 36853*	Duplicate of issettru 2816 and bj-issettruALTV 36855. Lemma for bj-denotesALTV 36854. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)
⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
Theorem	bj-denotesALTV 36854*	Moved to main as iseqsetv-clel 2817 and kept for the comments. 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 275 (to add an intermediate proposition ∃𝑧𝑧 = 𝐴 with a fresh 𝑧), cbvexvw 2033, and eqeq1 2738, requires the core axioms and { ax-9 2115, ax-ext 2705, df-cleq 2726 } whereas this proof requires the core axioms and { ax-8 2107, df-clab 2712, df-clel 2813 }. Theorem bj-issetwt 36857 proves that "existing" is equivalent to being a member of a class abstraction. It also requires, with the present proof, { ax-8 2107, df-clab 2712, df-clel 2813 } (whereas with the shorter proof from cbvexvw 2033 and eqeq1 2738 it would require { ax-8 2107, ax-9 2115, ax-ext 2705, df-clab 2712, df-cleq 2726, df-clel 2813 }). That every class is equal to a class abstraction is proved by abid1 2875, which requires { ax-8 2107, ax-9 2115, ax-ext 2705, df-clab 2712, df-cleq 2726, df-clel 2813 }. Note that there is no disjoint variable condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2374. Actually, the proof depends only on the logical axioms ax-1 6 through ax-7 2004 and sp 2180. 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 2705 and df-cleq 2726 (e.g., eqid 2734 and eqeq1 2738). In particular, one cannot even prove ⊢ ∃𝑥𝑥 = 𝐴 ⇒ ⊢ 𝐴 = 𝐴 without ax-ext 2705 and df-cleq 2726. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
Theorem	bj-issettruALTV 36855*	Moved to main as issettru 2816 and kept for the comments. Weak version of isset 3491 without ax-ext 2705. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)
⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
Theorem	bj-elabtru 36856	This is as close as we can get to proving extensionality for "the" "universal" class without ax-ext 2705. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ {𝑥 ∣ ⊤} ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
Theorem	bj-issetwt 36857*	Closed form of bj-issetw 36858. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
Theorem	bj-issetw 36858*	The closest one can get to isset 3491 without using ax-ext 2705. See also vexw 2717. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3491 using eleq2i 2830 (which requires ax-ext 2705 and df-cleq 2726). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)
Theorem	bj-issetiv 36859*	Version of bj-isseti 36860 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 3495 as long as elex 3498 is not available (and the non-dependence of bj-issetiv 36859 on special properties of the universal class V is obvious). Prefer its use over bj-isseti 36860 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 36860*	Version of isseti 3495 with a class variable 𝑉 in the hypothesis instead of V for extra generality. This is indeed more general than isseti 3495 as long as elex 3498 is not available (and the non-dependence of bj-isseti 36860 on special properties of the universal class V is obvious). Use bj-issetiv 36859 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 36861	A weak version of ralv 3505 not using ax-ext 2705 (nor df-cleq 2726, df-clel 2813, df-v 3479), and only core FOL axioms. See also bj-rexvw 36862. The analogues for reuv 3507 and rmov 3508 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ 𝜓    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥𝜑)
Theorem	bj-rexvw 36862	A weak version of rexv 3506 not using ax-ext 2705 (nor df-cleq 2726, df-clel 2813, df-v 3479), and only core FOL axioms. See also bj-ralvw 36861. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ 𝜓    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-rababw 36863	A weak version of rabab 3509 not using df-clel 2813 nor df-v 3479 (but requiring ax-ext 2705) nor ax-12 2174. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ 𝜓    ⇒   ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑}
Theorem	bj-rexcom4bv 36864*	Version of rexcom4b 3510 and bj-rexcom4b 36865 with a disjoint variable condition on 𝑥, 𝑉, hence removing dependency on df-sb 2062 and df-clab 2712 (so that it depends on df-clel 2813 and df-rex 3068 only on top of first-order logic). Prefer its use over bj-rexcom4b 36865 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 36865*	Remove from rexcom4b 3510 dependency on ax-ext 2705 and ax-13 2374 (and on df-or 848, df-cleq 2726, df-nfc 2889, df-v 3479). The hypothesis uses 𝑉 instead of V (see bj-isseti 36860 for the motivation). Use bj-rexcom4bv 36864 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 36866	The FOL content of ceqsalt 3512. Lemma for bj-ceqsalt 36868 and bj-ceqsaltv 36869. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜃 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥𝜃) → (∀𝑥(𝜃 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalt1 36867	The FOL content of ceqsalt 3512. Lemma for bj-ceqsalt 36868 and bj-ceqsaltv 36869. TODO: consider removing if it does not add anything to bj-ceqsalt0 36866. (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 36868*	Remove from ceqsalt 3512 dependency on ax-ext 2705 (and on df-cleq 2726 and df-v 3479). Note: this is not doable with ceqsralt 3513 (or ceqsralv 3519), which uses eleq1 2826, but the same dependence removal is possible for ceqsalg 3514, ceqsal 3516, ceqsalv 3518, cgsexg 3523, cgsex2g 3524, cgsex4g 3525, ceqsex 3527, ceqsexv 3529, ceqsex2 3534, ceqsex2v 3535, ceqsex3v 3536, ceqsex4v 3537, ceqsex6v 3538, ceqsex8v 3539, gencbvex 3540 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3541, gencbval 3542, vtoclgft 3551 (it uses Ⅎ, whose justification nfcjust 2888 does not use ax-ext 2705) and several other vtocl* theorems (see for instance bj-vtoclg1f 36900). See also bj-ceqsaltv 36869. (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 36869*	Version of bj-ceqsalt 36868 with a disjoint variable condition on 𝑥, 𝑉, removing dependency on df-sb 2062 and df-clab 2712. Prefer its use over bj-ceqsalt 36868 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 36870	The FOL content of ceqsalg 3514. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜒 → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalg 36871*	Remove from ceqsalg 3514 dependency on ax-ext 2705 (and on df-cleq 2726 and df-v 3479). See also bj-ceqsalgv 36873. (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 36872*	Alternate proof of bj-ceqsalg 36871. (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 36873*	Version of bj-ceqsalg 36871 with a disjoint variable condition on 𝑥, 𝑉, removing dependency on df-sb 2062 and df-clab 2712. Prefer its use over bj-ceqsalg 36871 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 36874*	Alternate proof of bj-ceqsalgv 36873. (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 36875*	Remove from ceqsal 3516 dependency on ax-ext 2705 (and on df-cleq 2726, df-v 3479, df-clab 2712, df-sb 2062). (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 36876*	Remove from ceqsalv 3518 dependency on ax-ext 2705 (and on df-cleq 2726, df-v 3479, df-clab 2712, df-sb 2062). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	bj-spcimdv 36877*	Remove from spcimdv 3592 dependency on ax-9 2115, ax-10 2138, ax-11 2154, ax-13 2374, ax-ext 2705, df-cleq 2726 (and df-nfc 2889, df-v 3479, df-or 848, df-tru 1539, df-nf 1780). For an even more economical version, see bj-spcimdvv 36878. (Contributed by BJ, 30-Nov-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-spcimdvv 36878*	Remove from spcimdv 3592 dependency on ax-7 2004, ax-8 2107, ax-10 2138, ax-11 2154, ax-12 2174 ax-13 2374, ax-ext 2705, df-cleq 2726, df-clab 2712 (and df-nfc 2889, df-v 3479, df-or 848, df-tru 1539, df-nf 1780) 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 36877. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
21.19.5.3  Characterization among sets versus among classes
Theorem	elelb 36879	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 36880	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))
21.19.5.4  The nonfreeness quantifier for classes In this section, we prove the symmetry of the nonfreeness quantifier for classes.
Theorem	bj-nfcsym 36881	The nonfreeness quantifier for classes defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 5380 with additional axioms; see also nfcv 2902). This could be proved from aecom 2429 and nfcvb 5381 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 2740 instead of equcomd 2015; removing dependency on ax-ext 2705 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2922, eleq2d 2824 (using elequ2 2120), nfcvf 2929, dvelimc 2928, dvelimdc 2927, nfcvf2 2930. (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)
21.19.5.5  Lemmas for class substitution Some useful theorems for dealing with substitutions: sbbi 2306, sbcbig 3845, sbcel1g 4421, sbcel2 4423, sbcel12 4416, sbceqg 4417, csbvarg 4439.
Theorem	bj-sbeqALT 36882*	Substitution in an equality (use the more general version bj-sbeq 36883 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 36883	Distribute proper substitution through an equality relation. (See sbceqg 4417). (Contributed by BJ, 6-Oct-2018.)
⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)
Theorem	bj-sbceqgALT 36884	Distribute proper substitution through an equality relation. Alternate proof of sbceqg 4417. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 4417, 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 36885*	Lemma for bj-csbsn 36886 (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 36886	Substitution in a singleton. (Contributed by BJ, 6-Oct-2018.)
⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}
Theorem	bj-sbel1 36887*	Version of sbcel1g 4421 when substituting a set. (Note: one could have a corresponding version of sbcel12 4416 when substituting a set, but the point here is that the antecedent of sbcel1g 4421 is not needed when substituting a set.) (Contributed by BJ, 6-Oct-2018.)
⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵)
Theorem	bj-abv 36888	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 36889	Alternate version of bj-abv 36888; shorter but uses ax-8 2107. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)
Theorem	bj-ab0 36890	The class of sets verifying a falsity is the empty set (closed form of abf 4411). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
Theorem	bj-abf 36891	Shorter proof of abf 4411 (which should be kept as abfALT). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ {𝑥 ∣ 𝜑} = ∅
Theorem	bj-csbprc 36892	More direct proof of csbprc 4414 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
21.19.5.6  Removing some axiom requirements and disjoint variable conditions
Theorem	bj-exlimvmpi 36893*	A Fol lemma (exlimiv 1927 followed by mpi 20). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpi 36894	Lemma for bj-vtoclg1f1 36899 (an instance of this lemma is a version of bj-vtoclg1f1 36899 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 36895	Lemma for theorems of the vtoclg 3553 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpbir 36896	Lemma for theorems of the vtoclg 3553 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ (∃𝑥𝜒 → 𝜑)
Theorem	bj-vtoclf 36897*	Remove dependency on ax-ext 2705, df-clab 2712 and df-cleq 2726 (and df-sb 2062 and df-v 3479) from vtoclf 3563. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ 𝑉    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	bj-vtocl 36898*	Remove dependency on ax-ext 2705, df-clab 2712 and df-cleq 2726 (and df-sb 2062 and df-v 3479) from vtocl 3557. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝑉    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	bj-vtoclg1f1 36899*	The FOL content of vtoclg1f 3569 (hence not using ax-ext 2705, df-cleq 2726, df-nfc 2889, df-v 3479). 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 2705; as a byproduct, this dispenses with ax-11 2154 and ax-13 2374). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑦 𝑦 = 𝐴 → 𝜓)
Theorem	bj-vtoclg1f 36900*	Reprove vtoclg1f 3569 from bj-vtoclg1f1 36899. This removes dependency on ax-ext 2705, df-cleq 2726 and df-v 3479. Use bj-vtoclg1fv 36901 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)