P. 378 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37701-37800   *Has distinct variable group(s)

Type	Label	Description
Statement
21.20.4.11  Really adding ax-12 The results in the previous section, as actually many theorems of the main part using ax-12 2212, actually only require sp 2219 (which is proved using ax-12 2212).
Theorem	bj-substax12 37701	Equivalent form of the axiom of substitution bj-ax12 37631. Although both sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 37662 on 𝑡, 𝜑) to hold, their equivalence holds without DV conditions. The forward implication is proved in modal (K4) while the reverse implication is proved in modal (T5). The LHS has the advantage of not involving nested quantifiers on the same variable. Its metaweakening is proved from the core axiom schemes in bj-substw 37702. In the LHS, the reverse implication holds by equs4 2445 (or equs4v 2032 if a DV condition is added on 𝑥, 𝑡 as in bj-ax12 37631), and the forward implication is sbalex 2278. The LHS can be read as saying that if there exists a variable equal to a given term witnessing a given formula, then all variables equal to that term also witness that formula. The equivalent form of the LHS using only primitive symbols is (∀𝑥(𝑥 = 𝑡 → 𝜑) ∨ ∀𝑥(𝑥 = 𝑡 → ¬ 𝜑)), which expresses that a given formula is true at all variables equal to a given term, or false at all these variables. An equivalent form of the LHS using only the existential quantifier is ¬ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑡 ∧ ¬ 𝜑)), which expresses that there can be no two variables both equal to a given term, one witnessing a formula and the other witnessing its negation. These equivalences do not hold in intuitionistic logic. The LHS should be the preferred form, and has the advantage of having no negation nor nested quantifiers. (Contributed by BJ, 21-May-2024.) (Proof modification is discouraged.)
⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
Theorem	bj-substw 37702*	Weak form of the LHS of bj-substax12 37701 proved from the core axiom schemes. Compare ax12w 2169. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))
21.20.4.12  Nonfreeness
Syntax	wnnf 37703	Syntax for the nonfreeness quantifier.
wff Ⅎ'𝑥𝜑
Definition	df-bj-nnf 37704	Definition of the nonfreeness quantifier. The formula Ⅎ'𝑥𝜑 has the intended meaning that the variable 𝑥 is semantically nonfree in the formula 𝜑. The motivation for this quantifier is to have a condition expressible in the logic which is as close as possible to the non-occurrence condition DV (𝑥, 𝜑) (in Metamath files, "$d x ph $."), which belongs to the metalogic. The standard syntactic nonfreeness condition, also expressed in the metalogic, is intermediate between these two notions: semantic nonfreeness implies syntactic nonfreeness, which implies non-occurrence. Both implications are strict; for the first, note that ⊢ Ⅎ'𝑥𝑥 = 𝑥, that is, 𝑥 is semantically (but not syntactically) nonfree in the formula 𝑥 = 𝑥; for the second, note that 𝑥 is syntactically nonfree in the formula ∀𝑥𝑥 = 𝑥 although it occurs in it. We now prove two metatheorems which make precise the above fact that, as far as proving power is concerned, the nonfreeness condition Ⅎ'𝑥𝜑 is very close to the non-occurrence condition DV (𝑥, 𝜑). Let S be a Metamath system with the FOL-syntax of (i)set.mm, containing intuitionistic positive propositional calculus and ax-5 1942 and ax5e 1944. Theorem 1. If the scheme (Ⅎ'𝑥𝜑 & PHI1 & ... & PHIn ⇒ PHI0, DV) is provable in S, then so is the scheme (PHI1 & ... & PHIn ⇒ PHI0, DV ∪ {{𝑥, 𝜑}}). Proof: By bj-nnfv 37745, we can prove (Ⅎ'𝑥𝜑, {{𝑥, 𝜑}}), from which the theorem follows. QED Theorem 2. Suppose that S also contains (the FOL version of) modal logic KB and commutation of quantifiers alcom 2195 and excom 2198 (possibly weakened by a DV condition on the quantifying variables), and that S can be axiomatized such that the only axioms with a DV condition involving a formula variable are among ax-5 1942, ax5e 1944, ax5ea 1945. If the scheme (PHI1 & ... & PHIn ⇒ PHI0, DV) is provable in S, then so is the scheme (Ⅎ'𝑥𝜑 & PHI1 & ... & PHIn ⇒ PHI0, DV ∖ {{𝑥, 𝜑}}). More precisely, if S contains modal 45 and if the variables quantified over in PHI0, ..., PHIn are among 𝑥1, ..., 𝑥m, then the scheme (PHI1 & ... & PHIn ⇒ (antecedent → PHI0), DV ∖ {{𝑥, 𝜑}}) is provable in S, where the antecedent is a finite conjunction of formulas of the form ∀𝑥i1 ...∀𝑥ip Ⅎ'𝑥𝜑 where the 𝑥ij's are among the 𝑥i's. Lemma: If 𝑥 ∉ OC(PHI), then S proves the scheme (Ⅎ'𝑥𝜑 ⇒ Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}). More precisely, if the variables quantified over in PHI are among 𝑥1, ..., 𝑥m, then ((antecedent → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) is provable in S, with the same form of antecedent as above. Proof: By induction on the height of PHI. We first note that by bj-nnfbi 37724 we can assume that PHI contains only primitive (as opposed to defined) symbols. For the base case, atomic formulas are either 𝜑, in which case the scheme to prove is an instance of id 23, or have variables all in OC(PHI) ∖ {𝜑}, so (Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) by bj-nnfv 37745, hence ((Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) by a1i 11. For the induction step, PHI is either an implication, a negation, a conjunction, a disjunction, a biconditional, a universal or an existential quantification of formulas where 𝑥 does not occur. We use respectively bj-nnfim 37729, bj-nnfnt 37727, bj-nnfan 37731, bj-nnfor 37733, bj-nnfbit 37735, bj-nnfalt 37767, bj-nnfext 37768. For instance, in the implication case, if we have by induction hypothesis ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) and ((∀𝑦1 ...∀𝑦n Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PSI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) ∖ {𝜑}}), then bj-nnfim 37729 yields (((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 ∧ ∀𝑦1 ...∀𝑦n Ⅎ'𝑥𝜑) → Ⅎ'𝑥 (PHI → PSI)), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI → PSI) ∖ {𝜑}}) and similarly for antecedents which are conjunctions as in the statement of the lemma. In the universal quantification case, say quantification over 𝑦, if we have by induction hypothesis ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}), then bj-nnfalt 37767 yields ((∀𝑦∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥∀𝑦 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(∀𝑦 PHI) ∖ {𝜑}}) and similarly for antecedents which are conjunctions as in the statement of the lemma. Note bj-nnfalt 37767 and bj-nnfext 37768 are proved from positive propositional calculus with alcom 2195 and excom 2198 (possibly weakened by a DV condition on the quantifying variables), and modalB (via bj-19.12 37700). QED Proof of the theorem: Consider a proof of that scheme directly from the axioms. Consider a step where a DV condition involving 𝜑 is used. By hypothesis, that step is an instance of ax-5 1942 or ax5e 1944 or ax5ea 1945. It has the form (PSI → ∀𝑥 PSI) where PSI has the form of the lemma and the DV conditions of the proof contain {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) }. Therefore, one has ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PSI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) ∖ {𝜑}}) for appropriate 𝑥i's, and by bj-nnfa 37705 we obtain ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → (PSI → ∀𝑥 PSI)), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) ∖ {𝜑}}) and similarly for antecedents which are conjunctions as in the statement of the theorem. Similarly if the step is using ax5e 1944 or ax5ea 1945, we would use bj-nnfe 37708 or bj-nnfea 37711 respectively. Therefore, taking as antecedent of the theorem to prove the conjunction of all the antecedents at each of these steps, we obtain a proof by "carrying the context over", which is possible, as in the deduction theorem when the step uses ax-mp 5, and when the step uses ax-gen 1827, by bj-nnf-alrim 37722 and bj-nnfa1 37761 (which requires modal 45). The condition DV (𝑥, 𝜑) is not required by the resulting proof. Finally, there may be in the global antecedent thus constructed some dummy variables, which can be removed by spvw 2013. QED Compared with df-nf 1816, the present definition is stricter on positive propositional calculus (bj-nnfnfTEMP 37717) and equivalent on core FOL plus sp 2219 (bj-nfnnfTEMP 37759). While being stricter, it still holds for non-occurring variables (bj-nnfv 37745), which is the basic requirement for this quantifier. In particular, it translates more closely the associated variable disjointness condition. Since the nonfreeness quantifier is a means to translate a variable disjointness condition from the metalogic to the logic, it seems preferable. Also, since nonfreeness is mainly used as a hypothesis, this definition would allow more theorems, notably the 19.xx theorems, to be proved from the core axioms, without needing a 19.xxv variant. One can devise infinitely many definitions increasingly close to the non-occurring condition, like ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ ∀𝑥((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ ∀𝑥∀𝑥... and each stronger definition would permit more theorems to be proved from the core axioms. A reasonable rule seems to be to stop before nested quantifiers appear (since they typically require ax-10 2177 to work with), and also not to have redundant conjuncts when full metacomplete FOL= is developed. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
Theorem	bj-nnfa 37705	Nonfreeness implies the equivalent of ax-5 1942. See nf5r 2230. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Theorem	bj-nnfad 37706	Nonfreeness implies the equivalent of ax-5 1942, deduction form. See nf5rd 2232. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	bj-nnfai 37707	Nonfreeness implies the equivalent of ax-5 1942, inference form. See nf5ri 2231. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	bj-nnfe 37708	Nonfreeness implies the equivalent of ax5e 1944. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
Theorem	bj-nnfed 37709	Nonfreeness implies the equivalent of ax5e 1944, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓))
Theorem	bj-nnfei 37710	Nonfreeness implies the equivalent of ax5e 1944, inference form. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → 𝜑)
Theorem	bj-nnfea 37711	Nonfreeness implies the equivalent of ax5ea 1945. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nnfead 37712	Nonfreeness implies the equivalent of ax5ea 1945, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
Theorem	bj-nnfeai 37713	Nonfreeness implies the equivalent of ax5ea 1945, inference form. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
Theorem	bj-alnnf 37714	In deduction-style proofs, it is equivalent to assert that the context holds for all values of a variable, or that is does not depend on that variable. (Contributed by BJ, 28-Mar-2026.)
⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → Ⅎ'𝑥𝜑))
Theorem	bj-alnnf2 37715	If a proposition holds, then it holds for all values of a given variable if and only if it does not depend on that variable. (Contributed by BJ, 28-Mar-2026.)
⊢ (𝜑 → (∀𝑥𝜑 ↔ Ⅎ'𝑥𝜑))
Theorem	bj-dfnnf2 37716	Alternate definition of df-bj-nnf 37704 using only primitive symbols (→, ¬, ∀) in each conjunct. (Contributed by BJ, 20-Aug-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ ((𝜑 → ∀𝑥𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
Theorem	bj-nnfnfTEMP 37717	New nonfreeness implies old nonfreeness on minimal implicational calculus (the proof indicates it uses ax-3 8 because of set.mm's definition of the biconditional, but the proof actually holds in minimal implicational calculus). (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1816 except via df-nf 1816 directly. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	bj-nnfim1 37718	A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
Theorem	bj-nnfim2 37719	A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → 𝜓)))
Theorem	bj-nnftht 37720	A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2219 (modal T), as in bj-nnfbi 37724. (Contributed by BJ, 28-Jul-2023.)
⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)
Theorem	bj-nnfth 37721	A variable is nonfree in a theorem, inference form. (Contributed by BJ, 28-Jul-2023.)
⊢ 𝜑    ⇒   ⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnf-alrim 37722	Proof of the closed form of alrimi 2249 from modalK (compare alrimiv 1959). See also bj-alrim 37670. Actually, most proofs between 19.3t 2237 and 2sbbid 2282 could be proved without ax-12 2212. (Contributed by BJ, 20-Aug-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-stdpc5t 37723	Alias of bj-nnf-alrim 37722 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2244 proved from modalK (obsoleting stdpc5v 1970). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 37722 instead. (New usaged is discouraged.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-nnfbi 37724	If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other. Compare nfbiit 1883. From this and bj-nnfim 37729 and bj-nnfnt 37727, one can prove analogous nonfreeness conservation results for other propositional operators. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 37720) in order not to require sp 2219 (modal T). (Contributed by BJ, 27-Aug-2023.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))
Theorem	bj-nnfbd0 37725	If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other, deduction form. The antecedent of the conclusion is in the "strong necessity" modality of modal logic (see also bj-nnftht 37720) in order not to require sp 2219 (modal T). See bj-nnfbi 37724. (Contributed by BJ, 21-Mar-2026.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ ∀𝑥𝜑) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
Theorem	bj-nnfbii 37726	If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi 37724. (Contributed by BJ, 18-Nov-2023.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)
Theorem	bj-nnfnt 37727	A variable is nonfree in a formula if and only if it is nonfree in its negation. The foward implication is intuitionistically valid (and that direction is sufficient for the purpose of recursively proving that some formulas have a given variable not free in them, like bj-nnfim 37729). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1888. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)
Theorem	bj-nnfnth 37728	A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.)
⊢ ¬ 𝜑    ⇒   ⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnfim 37729	Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication. (Contributed by BJ, 27-Aug-2023.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓))
Theorem	bj-nnfimd 37730	Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 → 𝜒))
Theorem	bj-nnfan 37731	Nonfreeness in both conjuncts implies nonfreeness in the conjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of conjunction in terms of implication and negation, so using bj-nnfim 37729, bj-nnfnt 37727 and bj-nnfbi 37724, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∧ 𝜓))
Theorem	bj-nnfand 37732	Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 37731, it has two more essential steps but fewer total steps (since there are fewer intermediate formulas to build) and is easier to follow and understand. This statement is of intermediate complexity: for simpler statements, closed-style proofs like that of bj-nnfan 37731 will generally be shorter than deduction-style proofs while still easy to follow, while for more complex statements, the opposite will be true (and deduction-style proofs like that of bj-nnfand 37732 will generally be easier to understand). (Contributed by BJ, 19-Nov-2023.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∧ 𝜒))
Theorem	bj-nnfor 37733	Nonfreeness in both disjuncts implies nonfreeness in the disjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of disjunction in terms of implication and negation, so using bj-nnfim 37729, bj-nnfnt 37727 and bj-nnfbi 37724, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∨ 𝜓))
Theorem	bj-nnford 37734	Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 37733 and bj-nnfand 37732. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒))
Theorem	bj-nnfbit 37735	Nonfreeness in both sides implies nonfreeness in the biconditional. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ↔ 𝜓))
Theorem	bj-nnfbid 37736	Nonfreeness in both sides implies nonfreeness in the biconditional, deduction form. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ↔ 𝜒))
Theorem	bj-nnf-exlim 37737	Proof of the closed form of exlimi 2253 from modalK (compare exlimiv 1962). See also bj-sylget2 37579. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))
Theorem	bj-19.21t 37738	Statement 19.21t 2242 proved from modalK (obsoleting 19.21v 1971). (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	bj-19.23t 37739	Statement 19.23t 2246 proved from modalK (obsoleting 19.23v 1974). (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	bj-19.36im 37740	One direction of 19.36 2266 from the same axioms as 19.36imv 1977. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-19.37im 37741	One direction of 19.37 2268 from the same axioms as 19.37imv 1979. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))
Theorem	bj-19.42t 37742	Closed form of 19.42 2272 from the same axioms as 19.42v 1985. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))
Theorem	bj-19.41t 37743	Closed form of 19.41 2271 from the same axioms as 19.41v 1981. The same is doable with 19.27 2263, 19.28 2264, 19.31 2270, 19.32 2269, 19.44 2273, 19.45 2274. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)))
Theorem	bj-pm11.53vw 37744	Version of pm11.53v 1976 with nonfreeness antecedents. One can also prove the theorem with antecedent (Ⅎ'𝑦∀𝑥𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓). (Contributed by BJ, 7-Oct-2024.)
⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-nnfv 37745*	A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.)
⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnfbd 37746*	If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi 37724. (Contributed by BJ, 27-Aug-2023.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
Theorem	bj-pm11.53a 37747*	A variant of pm11.53v 1976. One can similarly prove a variant with DV (𝑦, 𝜑) and ∀𝑦Ⅎ'𝑥𝜓 instead of DV (𝑥, 𝜓) and ∀𝑥Ⅎ'𝑦𝜑. (Contributed by BJ, 7-Oct-2024.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-equsvt 37748*	A variant of equsv 2035. (Contributed by BJ, 7-Oct-2024.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	bj-equsalvwd 37749*	Variant of equsalvw 2036. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
Theorem	bj-equsexvwd 37750*	Variant of equsexvw 2037. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒))
Theorem	bj-nnf-spim 37751*	A universal specialization result in deduction form, proved from ax-1 6 -- ax-6 1999, where the only DV condition is on 𝑥, 𝑦 and where 𝑥 should be nonfree in the new proposition 𝜒 (and in the context 𝜑). (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-nnf-spime 37752*	An existential generalization result in deduction form, from ax-1 6-- ax-6 1999, where the only DV condition is on 𝑥, 𝑦, and where 𝑥 should be nonfree in the new proposition 𝜒 (and in the context 𝜑). (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∃𝑥𝜒))
Theorem	bj-nnf-cbvaliv 37753*	The only DV conditions are those saying that 𝑦 is a fresh variable used to construct 𝜒. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	bj-sbievwd 37754*	Variant of sbievw 2130. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	bj-sbft 37755	Version of sbft 2303 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.)
⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑))
Theorem	bj-nnf-cbvali 37756*	Compared with bj-nnf-cbvaliv 37753, replacing the DV condition on 𝑦, 𝜓 with the nonfreeness condition requires ax-11 2193. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	bj-nnf-cbval 37757*	Compared with cbvalv1 2370, this saves ax-12 2212. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-dfnnf3 37758	Alternate definition of nonfreeness when sp 2219 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1816. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nfnnfTEMP 37759	New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2219. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1816 except via df-nf 1816 directly. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)
Theorem	bj-wnfnf 37760	When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). This could also be proved from bj-nnfim 37729, bj-nnfe1 37762 and bj-nnfa1 37761. (Contributed by BJ, 9-Dec-2023.)
⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
Theorem	bj-nnfa1 37761	See nfa1 2187. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ Ⅎ'𝑥∀𝑥𝜑
Theorem	bj-nnfe1 37762	See nfe1 2186. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ Ⅎ'𝑥∃𝑥𝜑
Theorem	bj-nnflemaa 37763	One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 37657. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜓))
Theorem	bj-nnflemee 37764	One of four lemmas for nonfreeness: antecedent and consequent both expressed using existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-nnflemae 37765	One of four lemmas for nonfreeness: antecedent expressed with universal quantifier and consequent expressed with existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜓))
Theorem	bj-nnflemea 37766	One of four lemmas for nonfreeness: antecedent expressed with existential quantifier and consequent expressed with universal quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-nnfalt 37767	See nfal 2353 and bj-nfalt 37690. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∀𝑥𝜑)
Theorem	bj-nnfext 37768	See nfex 2354 and bj-nfext 37691. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∃𝑥𝜑)
Theorem	bj-pm11.53v 37769	Version of pm11.53v 1976 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.)
⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
21.20.4.13  Adding ax-13
Theorem	bj-axc10 37770	Alternate proof of axc10 2414. Shorter. One can prove a version with DV (𝑥, 𝑦) without ax-13 2401, by using ax6ev 2001 instead of ax6e 2412. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-alequex 37771	A fol lemma. See alequexv 2033 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2415 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
Theorem	bj-spimt2 37772	A step in the proof of spimt 2415. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-cbv3ta 37773	Closed form of cbv3 2426. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-cbv3tb 37774	Closed form of cbv3 2426. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-hbsb3t 37775	A theorem close to a closed form of hbsb3 2516. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
Theorem	bj-hbsb3 37776	Shorter proof of hbsb3 2516. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1t 37777	A theorem close to a closed form of nfs1 2517. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1t2 37778	A theorem close to a closed form of nfs1 2517. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1 37779	Shorter proof of nfs1 2517 (three essential steps instead of four). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
21.20.4.14  Removing dependencies on ax-13 (and ax-11) It is known that ax-13 2401 is logically redundant (see ax13w 2172 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2401 from every theorem in set.mm which is totally unbundled (i.e., has disjoint variable conditions on all setvar variables). Indeed, start with the existing proof, and replace any occurrence of ax-13 2401 with ax13w 2172. This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2401 (and using ax6v 2000 / ax6ev 2001 instead of ax-6 1999 / ax6e 2412, as is currently done). One reason to be optimistic is that the first few utility theorems using ax-13 2401 (roughly 200 of them) are then used mainly with dummy variables, which one can assume distinct from any other, so that the unbundled versions of the utility theorems suffice. In this section, we prove versions of theorems in the main part with dv conditions and not requiring ax-13 2401, labeled bj-xxxv (we follow the proof of xxx but use ax6v 2000 and ax6ev 2001 instead of ax-6 1999 and ax6e 2412, and ax-5 1942 instead of ax13v 2402; shorter proofs may be possible). When no additional dv condition is required, we label it bj-xxx. It is important to keep all the bundled theorems already in set.mm, but one may also add the (partially) unbundled versions which dispense with ax-13 2401, so as to remove dependencies on ax-13 2401 from many existing theorems. UPDATE: it turns out that several theorems of the form bj-xxxv, or minor variations, are already in set.mm with label xxxw. It is also possible to remove dependencies on ax-11 2193, typically by replacing a nonfree hypothesis with a disjoint variable condition (see cbv3v2 2277 and following theorems).
Theorem	bj-axc10v 37780*	Version of axc10 2414 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-spimtv 37781*	Version of spimt 2415 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbv3hv2 37782*	Version of cbv3h 2433 with two disjoint variable conditions, which does not require ax-11 2193 nor ax-13 2401. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	bj-cbv1hv 37783*	Version of cbv1h 2434 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	bj-cbv2hv 37784*	Version of cbv2h 2435 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbv2v 37785*	Version of cbv2 2432 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvaldv 37786*	Version of cbvald 2436 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvexdv 37787*	Version of cbvexd 2437 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	bj-cbval2vv 37788*	Version of cbval2vv 2442 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
Theorem	bj-cbvex2vv 37789*	Version of cbvex2vv 2443 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
Theorem	bj-cbvaldvav 37790*	Version of cbvaldva 2438 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvexdvav 37791*	Version of cbvexdva 2439 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	bj-cbvex4vv 37792*	Version of cbvex4v 2444 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))    &   ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒)
Theorem	bj-equsalhv 37793*	Version of equsalh 2449 with a disjoint variable condition, which does not require ax-13 2401. Remark: this is the same as equsalhw 2324. TODO: delete after moving the following paragraph somewhere. Remarks: equsexvw 2037 has been moved to Main; Theorem ax13lem2 2405 has a DV version which is a simple consequence of ax5e 1944; Theorems nfeqf2 2406, dveeq2 2407, nfeqf1 2408, dveeq1 2409, nfeqf 2410, axc9 2411, ax13 2404, have dv versions which are simple consequences of ax-5 1942. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	bj-axc11nv 37794*	Version of axc11n 2455 with a disjoint variable condition; instance of aevlem 2089. TODO: delete after checking surrounding theorems. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	bj-aecomsv 37795*	Version of aecoms 2457 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2458 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5404). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	bj-axc11v 37796*	Version of axc11 2459 with a disjoint variable condition, which does not require ax-13 2401 nor ax-10 2177. Remark: the following theorems (hbae 2460, nfae 2462, hbnae 2461, nfnae 2463, hbnaes 2464) would need to be totally unbundled to be proved without ax-13 2401, hence would be simple consequences of ax-5 1942 or nfv 1946. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Theorem	bj-drnf2v 37797*	Version of drnf2 2473 with a disjoint variable condition, which does not require ax-10 2177, ax-11 2193, ax-12 2212, ax-13 2401. Instance of nfbidv 1954. The version of axc15 2451 with a disjoint variable condition is actually ax12v2 2214 (up to adding a superfluous antecedent). (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))
Theorem	bj-equs45fv 37798*	Version of equs45f 2488 with a disjoint variable condition, which does not require ax-13 2401. The version of equs5 2489 with a disjoint variable condition is actually sbalex 2278 (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	bj-hbs1 37799*	Version of hbsb2 2511 with a disjoint variable condition, which does not require ax-13 2401, and removal of ax-13 2401 from hbs1 2307. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1v 37800*	Version of nfsb2 2512 with a disjoint variable condition, which does not require ax-13 2401, and removal of ax-13 2401 from nfs1v 2192. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑