P. 375 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37401-37500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-nnfead 37401	Nonfreeness implies the equivalent of ax5ea 1946, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
Theorem	bj-nnfeai 37402	Nonfreeness implies the equivalent of ax5ea 1946, inference form. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
Theorem	bj-alnnf 37403	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 37404	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 37405	Alternate definition of df-bj-nnf 37393 using only primitive symbols (→, ¬, ∀) in each conjunct. (Contributed by BJ, 20-Aug-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ ((𝜑 → ∀𝑥𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
Theorem	bj-nnfnfTEMP 37406	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 1817 except via df-nf 1817 directly. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	bj-nnfim1 37407	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 37408	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 37409	A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2222 (modal T), as in bj-nnfbi 37413. (Contributed by BJ, 28-Jul-2023.)
⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)
Theorem	bj-nnfth 37410	A variable is nonfree in a theorem, inference form. (Contributed by BJ, 28-Jul-2023.)
⊢ 𝜑    ⇒   ⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnf-alrim 37411	Proof of the closed form of alrimi 2252 from modalK (compare alrimiv 1960). See also bj-alrim 37359. Actually, most proofs between 19.3t 2240 and 2sbbid 2286 could be proved without ax-12 2216. (Contributed by BJ, 20-Aug-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-stdpc5t 37412	Alias of bj-nnf-alrim 37411 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2247 proved from modalK (obsoleting stdpc5v 1971). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 37411 instead. (New usaged is discouraged.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-nnfbi 37413	If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other. Compare nfbiit 1884. From this and bj-nnfim 37418 and bj-nnfnt 37416, 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 37409) in order not to require sp 2222 (modal T). (Contributed by BJ, 27-Aug-2023.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))
Theorem	bj-nnfbd0 37414	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 37409) in order not to require sp 2222 (modal T). See bj-nnfbi 37413. (Contributed by BJ, 21-Mar-2026.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ ∀𝑥𝜑) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
Theorem	bj-nnfbii 37415	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 37413. (Contributed by BJ, 18-Nov-2023.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)
Theorem	bj-nnfnt 37416	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 37418). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1889. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)
Theorem	bj-nnfnth 37417	A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.)
⊢ ¬ 𝜑    ⇒   ⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnfim 37418	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 37419	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 37420	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 37418, bj-nnfnt 37416 and bj-nnfbi 37413, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∧ 𝜓))
Theorem	bj-nnfand 37421	Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 37420, 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 37420 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 37421 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 37422	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 37418, bj-nnfnt 37416 and bj-nnfbi 37413, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∨ 𝜓))
Theorem	bj-nnford 37423	Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 37422 and bj-nnfand 37421. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒))
Theorem	bj-nnfbit 37424	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 37425	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 37426	Proof of the closed form of exlimi 2256 from modalK (compare exlimiv 1963). See also bj-sylget2 37268. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))
Theorem	bj-19.21t 37427	Statement 19.21t 2245 proved from modalK (obsoleting 19.21v 1972). (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	bj-19.23t 37428	Statement 19.23t 2249 proved from modalK (obsoleting 19.23v 1975). (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	bj-19.36im 37429	One direction of 19.36 2269 from the same axioms as 19.36imv 1978. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-19.37im 37430	One direction of 19.37 2271 from the same axioms as 19.37imv 1980. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))
Theorem	bj-19.42t 37431	Closed form of 19.42 2275 from the same axioms as 19.42v 1986. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))
Theorem	bj-19.41t 37432	Closed form of 19.41 2274 from the same axioms as 19.41v 1982. The same is doable with 19.27 2266, 19.28 2267, 19.31 2273, 19.32 2272, 19.44 2276, 19.45 2277. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)))
Theorem	bj-pm11.53vw 37433	Version of pm11.53v 1977 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 37434*	A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.)
⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnfbd 37435*	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 37413. (Contributed by BJ, 27-Aug-2023.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
Theorem	bj-pm11.53a 37436*	A variant of pm11.53v 1977. 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 37437*	A variant of equsv 2036. (Contributed by BJ, 7-Oct-2024.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	bj-equsalvwd 37438*	Variant of equsalvw 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-equsexvwd 37439*	Variant of equsexvw 2038. (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 37440*	A universal specialization result in deduction form, proved from ax-1 6 -- ax-6 2000, 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 37441*	An existential generalization result in deduction form, from ax-1 6-- ax-6 2000, 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 37442*	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 37443*	Variant of sbievw 2131. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	bj-sbft 37444	Version of sbft 2308 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.)
⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑))
Theorem	bj-nnf-cbvali 37445*	Compared with bj-nnf-cbvaliv 37442, replacing the DV condition on 𝑦, 𝜓 with the nonfreeness condition requires ax-11 2195. (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 37446*	Compared with cbvalv1 2376, this saves ax-12 2216. (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 37447	Alternate definition of nonfreeness when sp 2222 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1817. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nfnnfTEMP 37448	New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2222. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1817 except via df-nf 1817 directly. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)
Theorem	bj-wnfnf 37449	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 37418, bj-nnfe1 37451 and bj-nnfa1 37450. (Contributed by BJ, 9-Dec-2023.)
⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
Theorem	bj-nnfa1 37450	See nfa1 2189. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ Ⅎ'𝑥∀𝑥𝜑
Theorem	bj-nnfe1 37451	See nfe1 2188. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ Ⅎ'𝑥∃𝑥𝜑
Theorem	bj-nnflemaa 37452	One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 37346. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜓))
Theorem	bj-nnflemee 37453	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 37454	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 37455	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 37456	See nfal 2359 and bj-nfalt 37379. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∀𝑥𝜑)
Theorem	bj-nnfext 37457	See nfex 2360 and bj-nfext 37380. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∃𝑥𝜑)
Theorem	bj-pm11.53v 37458	Version of pm11.53v 1977 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.)
⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
21.19.4.13  Adding ax-13
Theorem	bj-axc10 37459	Alternate proof of axc10 2420. Shorter. One can prove a version with DV (𝑥, 𝑦) without ax-13 2407, by using ax6ev 2002 instead of ax6e 2418. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-alequex 37460	A fol lemma. See alequexv 2034 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2421 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
Theorem	bj-spimt2 37461	A step in the proof of spimt 2421. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-cbv3ta 37462	Closed form of cbv3 2432. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-cbv3tb 37463	Closed form of cbv3 2432. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-hbsb3t 37464	A theorem close to a closed form of hbsb3 2522. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
Theorem	bj-hbsb3 37465	Shorter proof of hbsb3 2522. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1t 37466	A theorem close to a closed form of nfs1 2523. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1t2 37467	A theorem close to a closed form of nfs1 2523. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1 37468	Shorter proof of nfs1 2523 (three essential steps instead of four). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
21.19.4.14  Removing dependencies on ax-13 (and ax-11) It is known that ax-13 2407 is logically redundant (see ax13w 2174 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2407 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 2407 with ax13w 2174. This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2407 (and using ax6v 2001 / ax6ev 2002 instead of ax-6 2000 / ax6e 2418, as is currently done). One reason to be optimistic is that the first few utility theorems using ax-13 2407 (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 2407, labeled bj-xxxv (we follow the proof of xxx but use ax6v 2001 and ax6ev 2002 instead of ax-6 2000 and ax6e 2418, and ax-5 1943 instead of ax13v 2408; 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 2407, so as to remove dependencies on ax-13 2407 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 2195, typically by replacing a nonfree hypothesis with a disjoint variable condition (see cbv3v2 2280 and following theorems).
Theorem	bj-axc10v 37469*	Version of axc10 2420 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-spimtv 37470*	Version of spimt 2421 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbv3hv2 37471*	Version of cbv3h 2439 with two disjoint variable conditions, which does not require ax-11 2195 nor ax-13 2407. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	bj-cbv1hv 37472*	Version of cbv1h 2440 with a disjoint variable condition, which does not require ax-13 2407. (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 37473*	Version of cbv2h 2441 with a disjoint variable condition, which does not require ax-13 2407. (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 37474*	Version of cbv2 2438 with a disjoint variable condition, which does not require ax-13 2407. (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 37475*	Version of cbvald 2442 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvexdv 37476*	Version of cbvexd 2443 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	bj-cbval2vv 37477*	Version of cbval2vv 2448 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
Theorem	bj-cbvex2vv 37478*	Version of cbvex2vv 2449 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
Theorem	bj-cbvaldvav 37479*	Version of cbvaldva 2444 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvexdvav 37480*	Version of cbvexdva 2445 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	bj-cbvex4vv 37481*	Version of cbvex4v 2450 with a disjoint variable condition, which does not require ax-13 2407. (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 37482*	Version of equsalh 2455 with a disjoint variable condition, which does not require ax-13 2407. Remark: this is the same as equsalhw 2329. TODO: delete after moving the following paragraph somewhere. Remarks: equsexvw 2038 has been moved to Main; Theorem ax13lem2 2411 has a DV version which is a simple consequence of ax5e 1945; Theorems nfeqf2 2412, dveeq2 2413, nfeqf1 2414, dveeq1 2415, nfeqf 2416, axc9 2417, ax13 2410, have dv versions which are simple consequences of ax-5 1943. (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 37483*	Version of axc11n 2461 with a disjoint variable condition; instance of aevlem 2090. 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 37484*	Version of aecoms 2463 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2464 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5423). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	bj-axc11v 37485*	Version of axc11 2465 with a disjoint variable condition, which does not require ax-13 2407 nor ax-10 2179. Remark: the following theorems (hbae 2466, nfae 2468, hbnae 2467, nfnae 2469, hbnaes 2470) would need to be totally unbundled to be proved without ax-13 2407, hence would be simple consequences of ax-5 1943 or nfv 1947. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Theorem	bj-drnf2v 37486*	Version of drnf2 2479 with a disjoint variable condition, which does not require ax-10 2179, ax-11 2195, ax-12 2216, ax-13 2407. Instance of nfbidv 1955. Note that the version of axc15 2457 with a disjoint variable condition is actually ax12v2 2218 (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 37487*	Version of equs45f 2494 with a disjoint variable condition, which does not require ax-13 2407. Note that the version of equs5 2495 with a disjoint variable condition is actually sbalex 2281 (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 37488*	Version of hbsb2 2517 with a disjoint variable condition, which does not require ax-13 2407, and removal of ax-13 2407 from hbs1 2312. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1v 37489*	Version of nfsb2 2518 with a disjoint variable condition, which does not require ax-13 2407, and removal of ax-13 2407 from nfs1v 2194. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
Theorem	bj-hbsb2av 37490*	Version of hbsb2a 2519 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-hbsb3v 37491*	Version of hbsb3 2522 with a disjoint variable condition, which does not require ax-13 2407. (Remark: the unbundled version of nfs1 2523 is given by bj-nfs1v 37489.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfsab1 37492*	Remove dependency on ax-13 2407 from nfsab1 2752. UPDATE / TODO: nfsab1 2752 does not use ax-13 2407 either anymore; bj-nfsab1 37492 is shorter than nfsab1 2752 but uses ax-12 2216. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
Theorem	bj-dtrucor2v 37493*	Version of dtrucor2 5348 with a disjoint variable condition, which does not require ax-13 2407 (nor ax-4 1842, ax-5 1943, ax-7 2041, ax-12 2216). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)    ⇒   ⊢ (𝜑 ∧ ¬ 𝜑)
21.19.4.15  Distinct var metavariables The closed formula ∀𝑥∀𝑦𝑥 = 𝑦 approximately means that the var metavariables 𝑥 and 𝑦 represent the same variable vi. In a domain with at most one object, however, this formula is always true, hence the "approximately" in the previous sentence.
Theorem	bj-hbaeb2 37494	Biconditional version of a form of hbae 2466 with commuted quantifiers, not requiring ax-11 2195. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦)
Theorem	bj-hbaeb 37495	Biconditional version of hbae 2466. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)
Theorem	bj-hbnaeb 37496	Biconditional version of hbnae 2467 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	bj-dvv 37497	A special instance of bj-hbaeb2 37494. A lemma for distinct var metavariables. Note that the right-hand side is a closed formula (a sentence). (Contributed by BJ, 6-Oct-2018.)
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑦 𝑥 = 𝑦)
21.19.4.16  Around ~ equsal As a rule of thumb, if a theorem of the form ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (𝜒 ↔ 𝜃) is in the database, and the "more precise" theorems ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜒 → 𝜃) and ⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜃 → 𝜒) also hold (see bj-bisym 37224), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2453 (and equsalh 2455 and equsexh 2456). Even if only one of these two theorems holds, it should be added to the database.
Theorem	bj-equsal1t 37498	Duplication of wl-equsal1t 38238, with shorter proof. If one imposes a disjoint variable condition on 𝑥, 𝑦, then one can use alequexv 2034 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 38239 is also interesting. (Contributed by BJ, 6-Oct-2018.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	bj-equsal1ti 37499	Inference associated with bj-equsal1t 37498. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)
Theorem	bj-equsal1 37500	One direction of equsal 2452. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓)