P. 350 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34901-35000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-hbext 34901	Closed form of hbex 2320. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))
Theorem	bj-nfalt 34902	Closed form of nfal 2318. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)
Theorem	bj-nfext 34903	Closed form of nfex 2319. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)
Theorem	bj-eeanvw 34904*	Version of exdistrv 1960 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2155. (The same can be done with eeeanv 2349 and ee4anv 2350.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	bj-modal4 34905	First-order logic form of the modal axiom (4). See hba1 2291. This is the standard proof of the implication in modal logic (B5 ⇒ 4). Its dual statement is bj-modal4e 34906. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	bj-modal4e 34906	First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 34905 (hba1 2291). (Contributed by BJ, 21-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)
Theorem	bj-modalb 34907	A short form of the axiom B of modal logic using only primitive symbols (→ , ¬ , ∀). (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	bj-wnf1 34908	When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-wnf2 34909	When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-wnfanf 34910	When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the universal form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
Theorem	bj-wnfenf 34911	When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the existential form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → 𝜓))
Theorem	bj-substax12 34912	Equivalent form of the axiom of substitution bj-ax12 34847. Although both sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 34876 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 34913. Note that in the LHS, the reverse implication holds by equs4 2417 (or equs4v 2004 if a DV condition is added on 𝑥, 𝑡 as in bj-ax12 34847). The LHS can be read as saying that if there exists a setvar equal to a given term witnessing 𝜑, then all setvars equal to that term also witness 𝜑. An equivalent suggestive form for the LHS 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 𝜑 and the other witnessing ¬ 𝜑. (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 34913*	Weak form of the LHS of bj-substax12 34912 proved from the core axiom schemes. Compare ax12w 2130. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))
20.15.4.10  Nonfreeness
Syntax	wnnf 34914	Syntax for the nonfreeness quantifier.
wff Ⅎ'𝑥𝜑
Definition	df-bj-nnf 34915	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 1914 and ax5e 1916. 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 34945, 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 2157 and excom 2163 (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 1914, ax5e 1916, ax5ea 1917. 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 34916 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 22, or have variables all in OC(PHI) ∖ {𝜑}, so (Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) by bj-nnfv 34945, 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 34937, bj-nnfnt 34931, bj-nnfan 34939, bj-nnfor 34941, bj-nnfbit 34943, bj-nnfalt 34957, bj-nnfext 34958. 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 34937 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 34957 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 34957 and bj-nnfext 34958 are proved from positive propositional calculus with alcom 2157 and excom 2163 (possibly weakened by a DV condition on the quantifying variables), and modalB (via bj-19.12 34952). 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 1914 or ax5e 1916 or ax5ea 1917. 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 34919 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 1916 or ax5ea 1917, we would use bj-nnfe 34922 or bj-nnfea 34925 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 1798, by bj-nnf-alrim 34946 and bj-nnfa1 34950 (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 1985. QED Compared with df-nf 1787, the present definition is stricter on positive propositional calculus (bj-nnfnfTEMP 34929) and equivalent on core FOL plus sp 2177 (bj-nfnnfTEMP 34949). While being stricter, it still holds for non-occurring variables (bj-nnfv 34945), 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 2138 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-nnfbi 34916	If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other. Compare nfbiit 1854. From this and bj-nnfim 34937 and bj-nnfnt 34931, 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 34932) in order not to require sp 2177 (modal T). (Contributed by BJ, 27-Aug-2023.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))
Theorem	bj-nnfbd 34917*	If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi 34916. (Contributed by BJ, 27-Aug-2023.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
Theorem	bj-nnfbii 34918	If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi 34916. (Contributed by BJ, 18-Nov-2023.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)
Theorem	bj-nnfa 34919	Nonfreeness implies the equivalent of ax-5 1914. See nf5r 2188. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Theorem	bj-nnfad 34920	Nonfreeness implies the equivalent of ax-5 1914, deduction form. See nf5rd 2190. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	bj-nnfai 34921	Nonfreeness implies the equivalent of ax-5 1914, inference form. See nf5ri 2189. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	bj-nnfe 34922	Nonfreeness implies the equivalent of ax5e 1916. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
Theorem	bj-nnfed 34923	Nonfreeness implies the equivalent of ax5e 1916, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓))
Theorem	bj-nnfei 34924	Nonfreeness implies the equivalent of ax5e 1916, inference form. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → 𝜑)
Theorem	bj-nnfea 34925	Nonfreeness implies the equivalent of ax5ea 1917. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nnfead 34926	Nonfreeness implies the equivalent of ax5ea 1917, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
Theorem	bj-nnfeai 34927	Nonfreeness implies the equivalent of ax5ea 1917, inference form. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
Theorem	bj-dfnnf2 34928	Alternate definition of df-bj-nnf 34915 using only primitive symbols (→, ¬, ∀) in each conjunct. (Contributed by BJ, 20-Aug-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ ((𝜑 → ∀𝑥𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
Theorem	bj-nnfnfTEMP 34929	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 1787 except via df-nf 1787 directly. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	bj-wnfnf 34930	When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). Note that this could also be proved from bj-nnfim 34937, bj-nnfe1 34951 and bj-nnfa1 34950. (Contributed by BJ, 9-Dec-2023.)
⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
Theorem	bj-nnfnt 34931	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 34937). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1860. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)
Theorem	bj-nnftht 34932	A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2177 (modal T), as in bj-nnfbi 34916. (Contributed by BJ, 28-Jul-2023.)
⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)
Theorem	bj-nnfth 34933	A variable is nonfree in a theorem, inference form. (Contributed by BJ, 28-Jul-2023.)
⊢ 𝜑    ⇒   ⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnfnth 34934	A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.)
⊢ ¬ 𝜑    ⇒   ⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnfim1 34935	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 34936	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-nnfim 34937	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 34938	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 34939	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 34937, bj-nnfnt 34931 and bj-nnfbi 34916, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∧ 𝜓))
Theorem	bj-nnfand 34940	Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 34939, 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 34939 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 34940 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 34941	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 34937, bj-nnfnt 34931 and bj-nnfbi 34916, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∨ 𝜓))
Theorem	bj-nnford 34942	Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 34941 and bj-nnfand 34940. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒))
Theorem	bj-nnfbit 34943	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 34944	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-nnfv 34945*	A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.)
⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnf-alrim 34946	Proof of the closed form of alrimi 2207 from modalK (compare alrimiv 1931). See also bj-alrim 34884. Actually, most proofs between 19.3t 2195 and 2sbbid 2240 could be proved without ax-12 2172. (Contributed by BJ, 20-Aug-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-nnf-exlim 34947	Proof of the closed form of exlimi 2211 from modalK (compare exlimiv 1934). See also bj-sylget2 34812. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))
Theorem	bj-dfnnf3 34948	Alternate definition of nonfreeness when sp 2177 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1787. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nfnnfTEMP 34949	New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2177. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1787 except via df-nf 1787 directly. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)
Theorem	bj-nnfa1 34950	See nfa1 2149. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ Ⅎ'𝑥∀𝑥𝜑
Theorem	bj-nnfe1 34951	See nfe1 2148. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ Ⅎ'𝑥∃𝑥𝜑
Theorem	bj-19.12 34952	See 19.12 2322. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2163 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1787 or df-bj-nnf 34915, directly or indirectly. (Proof modification is discouraged.)
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)
Theorem	bj-nnflemaa 34953	One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 34872. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
Theorem	bj-nnflemee 34954	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 → 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑))
Theorem	bj-nnflemae 34955	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 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
Theorem	bj-nnflemea 34956	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 → 𝜑) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nnfalt 34957	See nfal 2318 and bj-nfalt 34902. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∀𝑥𝜑)
Theorem	bj-nnfext 34958	See nfex 2319 and bj-nfext 34903. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∃𝑥𝜑)
Theorem	bj-stdpc5t 34959	Alias of bj-nnf-alrim 34946 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2202 proved from modalK (obsoleting stdpc5v 1942). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 34946 instead. (New usaged is discouraged.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-19.21t 34960	Statement 19.21t 2200 proved from modalK (obsoleting 19.21v 1943). (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	bj-19.23t 34961	Statement 19.23t 2204 proved from modalK (obsoleting 19.23v 1946). (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	bj-19.36im 34962	One direction of 19.36 2224 from the same axioms as 19.36imv 1949. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-19.37im 34963	One direction of 19.37 2226 from the same axioms as 19.37imv 1952. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))
Theorem	bj-19.42t 34964	Closed form of 19.42 2230 from the same axioms as 19.42v 1958. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))
Theorem	bj-19.41t 34965	Closed form of 19.41 2229 from the same axioms as 19.41v 1954. The same is doable with 19.27 2221, 19.28 2222, 19.31 2228, 19.32 2227, 19.44 2231, 19.45 2232. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)))
Theorem	bj-sbft 34966	Version of sbft 2263 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.)
⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑))
Theorem	bj-pm11.53vw 34967	Version of pm11.53v 1948 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-pm11.53v 34968	Version of pm11.53v 1948 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.)
⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-pm11.53a 34969*	A variant of pm11.53v 1948. 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 34970*	A variant of equsv 2007. (Contributed by BJ, 7-Oct-2024.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	bj-equsalvwd 34971*	Variant of equsalvw 2008. (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 34972*	Variant of equsexvw 2009. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒))
Theorem	bj-sbievwd 34973*	Variant of sbievw 2096. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
20.15.4.11  Adding ax-13
Theorem	bj-axc10 34974	Alternate proof of axc10 2386. Shorter. One can prove a version with DV (𝑥, 𝑦) without ax-13 2373, by using ax6ev 1974 instead of ax6e 2384. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-alequex 34975	A fol lemma. See alequexv 2005 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2387 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
Theorem	bj-spimt2 34976	A step in the proof of spimt 2387. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-cbv3ta 34977	Closed form of cbv3 2398. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-cbv3tb 34978	Closed form of cbv3 2398. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-hbsb3t 34979	A theorem close to a closed form of hbsb3 2492. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
Theorem	bj-hbsb3 34980	Shorter proof of hbsb3 2492. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1t 34981	A theorem close to a closed form of nfs1 2493. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1t2 34982	A theorem close to a closed form of nfs1 2493. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1 34983	Shorter proof of nfs1 2493 (three essential steps instead of four). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
20.15.4.12  Removing dependencies on ax-13 (and ax-11) It is known that ax-13 2373 is logically redundant (see ax13w 2133 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2373 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 2373 with ax13w 2133. This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2373 (and using ax6v 1973 / ax6ev 1974 instead of ax-6 1972 / ax6e 2384, as is currently done). One reason to be optimistic is that the first few utility theorems using ax-13 2373 (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 2373, labeled bj-xxxv (we follow the proof of xxx but use ax6v 1973 and ax6ev 1974 instead of ax-6 1972 and ax6e 2384, and ax-5 1914 instead of ax13v 2374; 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 dipense with ax-13 2373, so as to remove dependencies on ax-13 2373 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 2155, typically by replacing a nonfree hypothesis with a disjoint variable condition (see cbv3v2 2235 and following theorems).
Theorem	bj-axc10v 34984*	Version of axc10 2386 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-spimtv 34985*	Version of spimt 2387 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbv3hv2 34986*	Version of cbv3h 2405 with two disjoint variable conditions, which does not require ax-11 2155 nor ax-13 2373. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	bj-cbv1hv 34987*	Version of cbv1h 2406 with a disjoint variable condition, which does not require ax-13 2373. (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 34988*	Version of cbv2h 2407 with a disjoint variable condition, which does not require ax-13 2373. (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 34989*	Version of cbv2 2404 with a disjoint variable condition, which does not require ax-13 2373. (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 34990*	Version of cbvald 2408 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvexdv 34991*	Version of cbvexd 2409 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	bj-cbval2vv 34992*	Version of cbval2vv 2414 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
Theorem	bj-cbvex2vv 34993*	Version of cbvex2vv 2415 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
Theorem	bj-cbvaldvav 34994*	Version of cbvaldva 2410 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvexdvav 34995*	Version of cbvexdva 2411 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	bj-cbvex4vv 34996*	Version of cbvex4v 2416 with a disjoint variable condition, which does not require ax-13 2373. (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 34997*	Version of equsalh 2421 with a disjoint variable condition, which does not require ax-13 2373. Remark: this is the same as equsalhw 2289. TODO: delete after moving the following paragraph somewhere. Remarks: equsexvw 2009 has been moved to Main; Theorem ax13lem2 2377 has a DV version which is a simple consequence of ax5e 1916; Theorems nfeqf2 2378, dveeq2 2379, nfeqf1 2380, dveeq1 2381, nfeqf 2382, axc9 2383, ax13 2376, have dv versions which are simple consequences of ax-5 1914. (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 34998*	Version of axc11n 2427 with a disjoint variable condition; instance of aevlem 2059. 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 34999*	Version of aecoms 2429 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2430 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5360). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	bj-axc11v 35000*	Version of axc11 2431 with a disjoint variable condition, which does not require ax-13 2373 nor ax-10 2138. Remark: the following theorems (hbae 2432, nfae 2434, hbnae 2433, nfnae 2435, hbnaes 2436) would need to be totally unbundled to be proved without ax-13 2373, hence would be simple consequences of ax-5 1914 or nfv 1918. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))