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Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-19.21t 32801 Proof of 19.21t 2072 from stdpc5t 32798. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 
Theoremexlimii 32802 Inference associated with exlimi 2085. 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 32803 Proof of ax-11 2033 similar to PM's proof of alcom 2036 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 32807. Axiom ax-11 2033 is used in the proof only through nfa2 2039. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremax6er 32804 Commuted form of ax6e 2249. (Could be placed right after ax6e 2249). (Contributed by BJ, 15-Sep-2018.)
𝑥 𝑦 = 𝑥
 
Theoremexlimiieq1 32805 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 32806 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 32807* Proof of ax-11 2033 from the standard axioms of predicate calculus, similar to PM's proof of alcom 2036 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 2033 is used in the proof only through nfal 2152, nfsb 2439, sbal 2461, sb8 2423. See also ax11-pm 32803. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
20.14.4.16  Alternate definition of substitution
 
Theorembj-sbsb 32808 Biconditional showing two possible (dual) definitions of substitution df-sb 1880 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
(((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜑)))
 
Theorembj-dfsb2 32809 Alternate (dual) definition of substitution df-sb 1880 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
([𝑦 / 𝑥]𝜑 ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜑)))
 
20.14.4.17  Lemmas for substitution
 
Theorembj-sbf3 32810 Substitution has no effect on a bound variabe (existential quantifier case); see sbf2 2381. (Contributed by BJ, 2-May-2019.)
([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-sbf4 32811 Substitution has no effect on a bound variabe (non-freeness case); see sbf2 2381. (Contributed by BJ, 2-May-2019.)
([𝑦 / 𝑥]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜑)
 
Theorembj-sbnf 32812* Move non-free predicate in and out of substitution; see sbal 2461 and sbex 2462. (Contributed by BJ, 2-May-2019.)
([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)
 
20.14.4.18  Existential uniqueness
 
Theorembj-eu3f 32813* Version of eu3v 2497 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 2497. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)
𝑦𝜑       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theorembj-eumo0 32814* Existential uniqueness implies "at most one." Used to be in the main part and deprecated in favor of eumo 2498 and mo2 2478. (Contributed by NM, 8-Jul-1994.) (Revised by BJ, 8-Jun-2019.)
𝑦𝜑       (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
20.14.4.19  First-logic: miscellaneous

Miscellaneous theorems of first-order logic.

 
Theorembj-sbidmOLD 32815 Obsolete proof of sbidm 2413 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 32816* Obsolete proof of mo3 2506 temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
 
Theorembj-syl66ib 32817 A mixed syllogism inference derived from syl6ib 241. In addition to bj-dvelimdv1 32819, it can also shorten alexsubALTlem4 21848 (4821>4812), supsrlem 9929 (2868>2863). (Contributed by BJ, 20-Oct-2021.)
(𝜑 → (𝜓𝜃))    &   (𝜃𝜏)    &   (𝜏𝜒)       (𝜑 → (𝜓𝜒))
 
Theorembj-dvelimdv 32818* Deduction form of dvelim 2336 with DV conditions. Uncurried (imported) form of bj-dvelimdv 32818. Typically, 𝑧 is a fresh variable used for the implicit substitution hypothesis that results in 𝜒 (namely, 𝜓 can be thought as 𝜓(𝑥, 𝑦) and 𝜒 as 𝜓(𝑥, 𝑧)). So the theorem says that if x is effectively free in 𝜓(𝑥, 𝑧), then if x and y are not the same variable, then 𝑥 is also effectively free in 𝜓(𝑥, 𝑦), in a context 𝜑.

One can weakend the implicit substitution hypothesis by adding the antecedent 𝜑 but this typically does not make the theorem much more useful. Similarly, one could use non-freeness hypotheses instead of DV conditions but since this result is typically used when 𝑧 is a dummy variable, this would not be of much benefit. One could also remove DV(z,x) since in the proof nfv 1842 can be replaced with nfal 2152 followed by nfn 1783.

Remark: nfald 2164 uses ax-11 2033; it might be possible to inline and use ax11w 2006 instead, but there is still a use via 19.12 2163 anyway. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝑧 = 𝑦 → (𝜒𝜓))       ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
 
Theorembj-dvelimdv1 32819* Curried (exported) form of bj-dvelimdv 32818. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝑧 = 𝑦 → (𝜒𝜓))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓))
 
Theorembj-dvelimv 32820* A version of dvelim 2336 using the "non-free" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑧 = 𝑦 → (𝜓𝜑))       (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
 
Theorembj-nfeel2 32821* Non-freeness in an equality. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝑧)
 
Theorembj-axc14nf 32822 Proof of a version of axc14 2371 using the "non-free" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥𝑦))
 
Theorembj-axc14 32823 Alternate proof of axc14 2371 (even when inlining the above results, this gives a shorter proof). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
 
20.14.5  Set theory
 
20.14.5.1  Eliminability of class terms

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

Eliminability of class variables using the $a-statements ax-ext 2601, df-clab 2608, df-cleq 2614, df-clel 2617 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 2601, df-clab 2608, df-cleq 2614, df-clel 2617 }) 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 32824 and the following theorems of this section, which are special instances of df-clab 2608, dfcleq 2615 (proved from {FOL, ax-ext 2601, df-cleq 2614 }), and df-clel 2617. 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 32825, eliminable2b 32826 and eliminable3a 32828, 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 1481, 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 2608 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 2608, ax-ext 2601 and df-cleq 2614 are sufficient (over FOL) to eliminate class terms.

To prove that { df-clab 2608, df-cleq 2614, df-clel 2617 } provides a definitional extension of {FOL, ax-ext 2601 }, 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 2608, df-cleq 2614, df-clel 2617 }. 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 2601 }. 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 2608, df-cleq 2614, df-clel 2617 }. It involves a careful case study on the structure of the proof tree.

 
Theoremeliminable1 32824 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 32825* 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 32826* 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 32827* 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 32828* 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 32829* 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 32830* Every class can be written as (is equal to) a class abstraction. cvjust 2616 is a special instance of it, but the present proof does not require ax-13 2245, contrary to cvjust 2616. This theorem requires ax-ext 2601, df-clab 2608, df-cleq 2614, df-clel 2617, 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 1481, cab 2607 and statements corresponding to defined class constructors.

UPDATE: This theorem is (almost) abid2 2744 and bj-abid2 32766, though the present proof is shorter than a proof from bj-abid2 32766 and eqcomi 2630 (and is shorter than the proof of either); plus, it is of the same form as cvjust 2616 and such a basic statement deserves to be present in both forms. Note that bj-termab 32830 shortens the proof of abid2 2744, and shortens five proofs by a total of 72 bytes. Move it to Main as "abid1" proved from abbi2i 2737? Note also that this is the form in Quine, more than abid2 2744. (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 2601. One could move all theorems from cab 2607 to df-clel 2617 (except for dfcleq 2615 and cvjust 2616) 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 2614.

Note that without ax-ext 2601, the $a-statements df-clab 2608, df-cleq 2614, and df-clel 2617 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 2601, 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 1989, wel 1990, ax-8 1991, ax-9 1998).

Remark: the weakening of eleq1 2688 / eleq2 2689 to eleq1w 2683 / eleq2w 2684 can also be done with eleq1i 2691, eqeltri 2696, eqeltrri 2697, eleq1a 2695, eleq1d 2685, eqeltrd 2700, eqeltrrd 2701, eqneltrd 2719, eqneltrrd 2720, nelneq 2724.

 
Theorembj-cleljustab 32831* An instance of df-clel 2617 where the LHS (the definiendum) has the form "setvar class abstraction". The straightforward yet important fact that this statement can be proved from FOL= and df-clab 2608 (hence without df-clel 2617 or df-cleq 2614) was stressed by Mario Carneiro. The instance of df-clel 2617 where the LHS has the form "setvar setvar" is proved as cleljust 1997, from FOL= and ax-8 1991. Note: when df-ssb 32604 is the official definition for substitution, one can use bj-ssbequ instead of sbequ 2375 to prove bj-cleljustab 32831 from Tarski's FOL= with df-clab 2608. (Contributed by BJ, 8-Nov-2021.) (Proof modification is discouraged.)
(𝑥 ∈ {𝑦𝜑} ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
 
Theorembj-clelsb3 32832* Remove dependency on ax-ext 2601 (and df-cleq 2614) from clelsb3 2728. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
 
Theorembj-hblem 32833* Remove dependency on ax-ext 2601 (and df-cleq 2614) from hblem 2730. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝑧𝐴 → ∀𝑥 𝑧𝐴)
 
Theorembj-nfcjust 32834* Remove dependency on ax-ext 2601 (and df-cleq 2614 and ax-13 2245) from nfcjust 2751. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
 
Theorembj-nfcrii 32835* Remove dependency on ax-ext 2601 (and df-cleq 2614) from nfcrii 2756. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 
Theorembj-nfcri 32836* Remove dependency on ax-ext 2601 (and df-cleq 2614) from nfcri 2757. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝐴       𝑥 𝑦𝐴
 
Theorembj-nfnfc 32837 Remove dependency on ax-ext 2601 (and df-cleq 2614) from nfnfc 2773. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝐴       𝑥𝑦𝐴
 
Theorembj-vexwt 32838 Closed form of bj-vexw 32839. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) Use bj-vexwvt 32840 instead when sufficient. (New usage is discouraged.)
(∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
 
Theorembj-vexw 32839 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 3201, without using ax-ext 2601. Note that this theorem has no dv condition and does not use df-clel 2617 nor df-cleq 2614 either: only first-order logic and df-clab 2608.

Without ax-ext 2601, one cannot define "the" universal class, since one could not prove for instance the justification theorem {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} (see vjust 3199). Indeed, in order to prove any equality of classes, one needs df-cleq 2614, which has ax-ext 2601 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 2601. See also bj-issetw 32844.

A version with a dv condition between 𝑥 and 𝑦 and not requiring ax-13 2245 is proved as bj-vexwv 32841, while the degenerate instance is a simple consequence of abid 2609. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.) Use bj-vexwv 32841 instead when sufficient. (New usage is discouraged.)

𝜑       𝑦 ∈ {𝑥𝜑}
 
Theorembj-vexwvt 32840* Closed form of bj-vexwv 32841 and version of bj-vexwt 32838 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
 
Theorembj-vexwv 32841* Version of bj-vexw 32839 with a dv condition, which does not require ax-13 2245. The degenerate instance of bj-vexw 32839 is a simple consequence of abid 2609 (which does not depend on ax-13 2245 either). (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
𝜑       𝑦 ∈ {𝑥𝜑}
 
Theorembj-denotes 32842* 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 2601 nor df-cleq 2614 (but of course using df-clab 2608 and df-clel 2617). Once extensionality is postulated, then isset 3205 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 2245. Actually, the proof depends only on ax-1--7 and sp 2052.

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 2601 (e.g., eqid 2621). In particular, one cannot even prove 𝑥𝑥 = 𝐴𝐴 = 𝐴.

With ax-ext 2601, the present theorem is obvious from cbvexv 2274 and eqeq1 2625 (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 32843* Closed form of bj-issetw 32844. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
(∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
 
Theorembj-issetw 32844* The closest one can get to isset 3205 without using ax-ext 2601. See also bj-vexw 32839. 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 32845* Version of bj-elisset 32846 with a dv condition on 𝑥, 𝑉. This proof uses only df-ex 1704, ax-gen 1721, ax-4 1736 and df-clel 2617 on top of propositional calculus. Prefer its use over bj-elisset 32846 when sufficient. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
 
Theorembj-elisset 32846* Remove from elisset 3213 dependency on ax-ext 2601 (and on df-cleq 2614 and df-v 3200). This proof uses only df-clab 2608 and df-clel 2617 on top of first-order logic. It only requires ax-1--7 and sp 2052. Use bj-elissetv 32845 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 32847* Version of bj-isseti 32848 with a dv condition on 𝑥, 𝑉. This proof uses only df-ex 1704, ax-gen 1721, ax-4 1736 and df-clel 2617 on top of propositional calculus. Prefer its use over bj-isseti 32848 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 32848* Remove from isseti 3207 dependency on ax-ext 2601 (and on df-cleq 2614 and df-v 3200). This proof uses only df-clab 2608 and df-clel 2617 on top of first-order logic. It only uses ax-12 2046 among the auxiliary logical axioms. The hypothesis uses 𝑉 instead of V for extra generality. This is indeed more general as long as elex 3210 is not available. Use bj-issetiv 32847 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 32849 A weak version of ralv 3217 not using ax-ext 2601 (nor df-cleq 2614, df-clel 2617, df-v 3200), but using ax-13 2245. For the sake of illustration, the next theorem bj-rexvwv 32850, a weak version of rexv 3218, has a dv condition and avoids dependency on ax-13 2245, while the analogues for reuv 3219 and rmov 3220 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝜓       (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥𝜑)
 
Theorembj-rexvwv 32850* A weak version of rexv 3218 not using ax-ext 2601 (nor df-cleq 2614, df-clel 2617, df-v 3200) with an additional dv condition to avoid dependency on ax-13 2245 as well. See bj-ralvw 32849. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝜓       (∃𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∃𝑥𝜑)
 
Theorembj-rababwv 32851* A weak version of rabab 3221 not using df-clel 2617 nor df-v 3200 (but requiring ax-ext 2601). A version without dv condition is provable by replacing bj-vexwv 32841 with bj-vexw 32839 in the proof, hence requiring ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝜓       {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥𝜑}
 
Theorembj-ralcom4 32852* Remove from ralcom4 3222 dependency on ax-ext 2601 and ax-13 2245 (and on df-or 385, df-an 386, df-tru 1485, df-sb 1880, df-clab 2608, df-cleq 2614, df-clel 2617, df-nfc 2752, df-v 3200). This proof uses only df-ral 2916 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
 
Theorembj-rexcom4 32853* Remove from rexcom4 3223 dependency on ax-ext 2601 and ax-13 2245 (and on df-or 385, df-tru 1485, df-sb 1880, df-clab 2608, df-cleq 2614, df-clel 2617, df-nfc 2752, df-v 3200). This proof uses only df-rex 2917 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
(∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
 
Theorembj-rexcom4a 32854* Remove from rexcom4a 3224 dependency on ax-ext 2601 and ax-13 2245 (and on df-or 385, df-sb 1880, df-clab 2608, df-cleq 2614, df-clel 2617, df-nfc 2752, df-v 3200). This proof uses only df-rex 2917 on top of first-order logic. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
(∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
 
Theorembj-rexcom4bv 32855* Version of bj-rexcom4b 32856 with a dv condition on 𝑥, 𝑉, hence removing dependency on df-sb 1880 and df-clab 2608 (so that it depends on df-clel 2617 and df-rex 2917 only on top of first-order logic). Prefer its use over bj-rexcom4b 32856 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
𝐵𝑉       (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
 
Theorembj-rexcom4b 32856* Remove from rexcom4b 3225 dependency on ax-ext 2601 and ax-13 2245 (and on df-or 385, df-cleq 2614, df-nfc 2752, df-v 3200). The hypothesis uses 𝑉 instead of V (see bj-isseti 32848 for the motivation). Use bj-rexcom4bv 32855 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝐵𝑉       (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
 
Theorembj-ceqsalt0 32857 The FOL content of ceqsalt 3226. Lemma for bj-ceqsalt 32859 and bj-ceqsaltv 32860. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜃 → (𝜑𝜓)) ∧ ∃𝑥𝜃) → (∀𝑥(𝜃𝜑) ↔ 𝜓))
 
Theorembj-ceqsalt1 32858 The FOL content of ceqsalt 3226. Lemma for bj-ceqsalt 32859 and bj-ceqsaltv 32860. (TODO: consider removing if it does not add anything to bj-ceqsalt0 32857.) (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
(𝜃 → ∃𝑥𝜒)       ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) ↔ 𝜓))
 
Theorembj-ceqsalt 32859* Remove from ceqsalt 3226 dependency on ax-ext 2601 (and on df-cleq 2614 and df-v 3200). Note: this is not doable with ceqsralt 3227 (or ceqsralv 3232), which uses eleq1 2688, but the same dependence removal is possible for ceqsalg 3228, ceqsal 3230, ceqsalv 3231, cgsexg 3236, cgsex2g 3237, cgsex4g 3238, ceqsex 3239, ceqsexv 3240, ceqsex2 3242, ceqsex2v 3243, ceqsex3v 3244, ceqsex4v 3245, ceqsex6v 3246, ceqsex8v 3247, gencbvex 3248 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3249, gencbval 3250, vtoclgft 3252 (it uses , whose justification nfcjust 2751 is actually provable without ax-ext 2601, as bj-nfcjust 32834 shows) and several other vtocl* theorems (see for instance bj-vtoclg1f 32895). See also bj-ceqsaltv 32860. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theorembj-ceqsaltv 32860* Version of bj-ceqsalt 32859 with a dv condition on 𝑥, 𝑉, removing dependency on df-sb 1880 and df-clab 2608. Prefer its use over bj-ceqsalt 32859 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theorembj-ceqsalg0 32861 The FOL content of ceqsalg 3228. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒 → (𝜑𝜓))       (∃𝑥𝜒 → (∀𝑥(𝜒𝜑) ↔ 𝜓))
 
Theorembj-ceqsalg 32862* Remove from ceqsalg 3228 dependency on ax-ext 2601 (and on df-cleq 2614 and df-v 3200). See also bj-ceqsalgv 32864. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theorembj-ceqsalgALT 32863* Alternate proof of bj-ceqsalg 32862. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theorembj-ceqsalgv 32864* Version of bj-ceqsalg 32862 with a dv condition on 𝑥, 𝑉, removing dependency on df-sb 1880 and df-clab 2608. Prefer its use over bj-ceqsalg 32862 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theorembj-ceqsalgvALT 32865* Alternate proof of bj-ceqsalgv 32864. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theorembj-ceqsal 32866* Remove from ceqsal 3230 dependency on ax-ext 2601 (and on df-cleq 2614, df-v 3200, df-clab 2608, df-sb 1880). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 
Theorembj-ceqsalv 32867* Remove from ceqsalv 3231 dependency on ax-ext 2601 (and on df-cleq 2614, df-v 3200, df-clab 2608, df-sb 1880). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 
Theorembj-spcimdv 32868* Remove from spcimdv 3288 dependency on ax-9 1998, ax-10 2018, ax-11 2033, ax-13 2245, ax-ext 2601, df-cleq 2614 (and df-nfc 2752, df-v 3200, df-or 385, df-tru 1485, df-nf 1709). For an even more economical version, see bj-spcimdvv 32869. (Contributed by BJ, 30-Nov-2020.) (Proof modification is discouraged.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theorembj-spcimdvv 32869* Remove from spcimdv 3288 dependency on ax-7 1934, ax-8 1991, ax-10 2018, ax-11 2033, ax-12 2046 ax-13 2245, ax-ext 2601, df-cleq 2614, df-clab 2608 (and df-nfc 2752, df-v 3200, df-or 385, df-tru 1485, df-nf 1709) at the price of adding a DV condition on 𝑥, 𝐵 (but in usages, 𝑥 is typically a dummy, hence fresh, variable). For the version without this DV condition, see bj-spcimdv 32868. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
20.14.5.3  The class-form not-free predicate

In this section, we prove the symmetry of the class-form not-free predicate.

 
Theorembj-nfcsym 32870 The class-form not-free predicate defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 4895 with additional axioms; see also nfcv 2763). This could be proved from aecom 2310 and nfcvb 4896 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 2627 instead of equcomd 1945; removing dependency on ax-ext 2601 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2781, eleq2d 2686 (using elequ2 2003), nfcvf 2787, dvelimc 2786, dvelimdc 2785, nfcvf2 2788. (Proof modification is discouraged.)
(𝑥𝑦𝑦𝑥)
 
20.14.5.4  Proposal for the definitions of class membership and class equality

In this section, we show (bj-ax8 32871 and bj-ax9 32874) that the current forms of the definitions of class membership (df-clel 2617) and class equality (df-cleq 2614) are too powerful, and we propose alternate definitions (bj-df-clel 32872 and bj-df-cleq 32877) which also have the advantage of making it clear that these definitions are conservative.

 
Theorembj-ax8 32871 Proof of ax-8 1991 from df-clel 2617 (and FOL). This shows that df-clel 2617 is "too powerful". A possible definition is given by bj-df-clel 32872. (Contributed by BJ, 27-Jun-2019.) Also a direct consequence of eleq1w 2683, which has essentially the same proof. (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theorembj-df-clel 32872* Candidate definition for df-clel 2617 (the need for it is exposed in bj-ax8 32871). The similarity of the hypothesis and the conclusion, together with all possible dv conditions, makes it clear that this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This definition should be directly referenced only by bj-dfclel 32873, which should be used instead. The proof is irrelevant since this is a proposal for an axiom.

Note: the current definition df-clel 2617 already mentions cleljust 1997 as a justification; here, we merely propose to put it (more preciesly: its universal closure) as a hypothesis to make things more explicit. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)

𝑢𝑣(𝑢𝑣 ↔ ∃𝑤(𝑤 = 𝑢𝑤𝑣))       (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
 
Theorembj-dfclel 32873* Characterization of the elements of a class. Note: cleljust 1997 could be relabeled "clelhyp". (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)
(𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
 
Theorembj-ax9 32874* Proof of ax-9 1998 from Tarski's FOL=, sp 2052, df-cleq 2614 and ax-ext 2601 (with two extra dv conditions on 𝑥, 𝑧 and 𝑦, 𝑧). For a version without these dv conditions, see bj-ax9-2 32875. This shows that df-cleq 2614 is "too powerful". A possible definition is given by bj-df-cleq 32877. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theorembj-ax9-2 32875 Proof of ax-9 1998 from Tarski's FOL=, ax-8 1991 (specifically, ax8v1 1993 and ax8v2 1994) , df-cleq 2614 and ax-ext 2601. For a version not using ax-8 1991, see bj-ax9 32874. This shows that df-cleq 2614 is "too powerful". A possible definition is given by bj-df-cleq 32877. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theorembj-cleqhyp 32876* The hypothesis of bj-df-cleq 32877. Note that the hypothesis of bj-df-cleq 32877 actually has an additional dv condition on 𝑥, 𝑦 and therefore is provable by simply using ax-ext 2601 in place of axext3 2603 in the current proof. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
 
Theorembj-df-cleq 32877* Candidate definition for df-cleq 2614 (the need for it is exposed in bj-ax9 32874). The similarity of the hypothesis and the conclusion makes it clear that this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This definition should be directly referenced only by bj-dfcleq 32878, which should be used instead. The proof is irrelevant since this is a proposal for an axiom.

Another definition, which would give finer control, is actually the pair of definitions where each has one implication of the present biconditional as hypothesis and conclusion. They assert that extensionality (respectively, the left-substitution axiom for the membership predicate) extends from setvars to classes. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

𝑢𝑣(𝑢 = 𝑣 ↔ ∀𝑤(𝑤𝑢𝑤𝑣))       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theorembj-dfcleq 32878* Proof of class extensionality from the axiom of set extensionality (ax-ext 2601) and the definition of class equality (bj-df-cleq 32877). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
20.14.5.5  Lemmas for class substitution

Some useful theorems for dealing with substitutions: sbbi 2400, sbcbig 3478, sbcel1g 3985, sbcel2 3987, sbcel12 3981, sbceqg 3982, csbvarg 4001.

 
Theorembj-sbeqALT 32879* Substitution in an equality (use the more genereal version bj-sbeq 32880 instead, without disjoint variable condition). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)
 
Theorembj-sbeq 32880 Distribute proper substitution through an equality relation. (See sbceqg 3982). (Contributed by BJ, 6-Oct-2018.)
([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)
 
Theorembj-sbceqgALT 32881 Distribute proper substitution through an equality relation. Alternate proof of sbceqg 3982. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 3982, but "minimize */except sbceqg" is ok. (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
 
Theorembj-csbsnlem 32882* Lemma for bj-csbsn 32883 (in this lemma, 𝑥 cannot occur in 𝐴). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
𝐴 / 𝑥{𝑥} = {𝐴}
 
Theorembj-csbsn 32883 Substitution in a singleton. (Contributed by BJ, 6-Oct-2018.)
𝐴 / 𝑥{𝑥} = {𝐴}
 
Theorembj-sbel1 32884* Version of sbcel1g 3985 when substituting a set. (Note: one could have a corresponding version of sbcel12 3981 when substituting a set, but the point here is that the antecedent of sbcel1g 3985 is not needed when substituting a set.) (Contributed by BJ, 6-Oct-2018.)
([𝑦 / 𝑥]𝐴𝐵𝑦 / 𝑥𝐴𝐵)
 
Theorembj-abv 32885 The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
(∀𝑥𝜑 → {𝑥𝜑} = V)
 
Theorembj-ab0 32886 The class of sets verifying a falsity is the empty set (closed form of abf 3976). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
(∀𝑥 ¬ 𝜑 → {𝑥𝜑} = ∅)
 
Theorembj-abf 32887 Shorter proof of abf 3976 (which should be kept as abfALT). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
¬ 𝜑       {𝑥𝜑} = ∅
 
Theorembj-csbprc 32888 More direct proof of csbprc 3978 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
 
20.14.5.6  Removing some dv conditions
 
Theorembj-exlimmpi 32889 Lemma for bj-vtoclg1f1 32894 (an instance of this lemma is a version of bj-vtoclg1f1 32894 where 𝑥 and 𝑦 are identified). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒 → (𝜑𝜓))    &   𝜑       (∃𝑥𝜒𝜓)
 
Theorembj-exlimmpbi 32890 Lemma for theorems of the vtoclg 3264 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒 → (𝜑𝜓))    &   𝜑       (∃𝑥𝜒𝜓)
 
Theorembj-exlimmpbir 32891 Lemma for theorems of the vtoclg 3264 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜒 → (𝜑𝜓))    &   𝜓       (∃𝑥𝜒𝜑)
 
Theorembj-vtoclf 32892* Remove dependency on ax-ext 2601, df-clab 2608 and df-cleq 2614 (and df-sb 1880 and df-v 3200) from vtoclf 3256. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   𝐴𝑉    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓
 
Theorembj-vtocl 32893* Remove dependency on ax-ext 2601, df-clab 2608 and df-cleq 2614 (and df-sb 1880 and df-v 3200) from vtocl 3257. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝐴𝑉    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓
 
Theorembj-vtoclg1f1 32894* The FOL content of vtoclg1f 3263 (hence not using ax-ext 2601, df-cleq 2614, df-nfc 2752, df-v 3200). Note the weakened "major" hypothesis and the dv condition between 𝑥 and 𝐴 (needed since the class-form non-free predicate is not available without ax-ext 2601; as a byproduct, this dispenses with ax-11 2033 and ax-13 2245). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (∃𝑦 𝑦 = 𝐴𝜓)
 
Theorembj-vtoclg1f 32895* Reprove vtoclg1f 3263 from bj-vtoclg1f1 32894. This removes dependency on ax-ext 2601, df-cleq 2614 and df-v 3200. Use bj-vtoclg1fv 32896 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)
 
Theorembj-vtoclg1fv 32896* Version of bj-vtoclg1f 32895 with a dv condition on 𝑥, 𝑉. This removes dependency on df-sb 1880 and df-clab 2608. Prefer its use over bj-vtoclg1f 32895 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)
 
Theorembj-rabbida2 32897 Version of rabbidva2 3184 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theorembj-rabbida 32898 Version of rabbidva 3186 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theorembj-rabbid 32899 Version of rabbidv 3187 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theorembj-rabeqd 32900 Deduction form of rabeq 3190. Note that contrary to rabeq 3190 it has no dv condition. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
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