P. 367 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36601-36700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-2albi 36601	Closed form of 2albii 1820. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))
Theorem	bj-notalbii 36602	Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4343 (103>94), ballotlem2 34480 (2655>2648), bnj1143 34780 (522>519), hausdiag 23532 (2119>2104). (Contributed by BJ, 17-Jul-2021.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)
Theorem	bj-2exbi 36603	Closed form of 2exbii 1849. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓))
Theorem	bj-3exbi 36604	Closed form of 3exbii 1850. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓))
Theorem	bj-sylggt 36605	Stronger form of sylgt 1822, closer to ax-2 7. (Contributed by BJ, 30-Jul-2025.)
⊢ ((𝜑 → ∀𝑥(𝜓 → 𝜒)) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))
Theorem	bj-sylgt2 36606	Uncurried (imported) form of sylgt 1822. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜓 → 𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))
Theorem	bj-alrimg 36607	The general form of the *alrim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in an antecedent, one can deduce from the universally quantified implication an implication where the consequent is universally quantified. Dual of bj-exlimg 36611. (Contributed by BJ, 9-Dec-2023.)
⊢ ((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓 → 𝜒) → (𝜑 → ∀𝑥𝜒)))
Theorem	bj-alrimd 36608	A slightly more general alrimd 2216. A common usage will have 𝜑 substituted for 𝜓 and 𝜒 substituted for 𝜃, giving a form closer to alrimd 2216. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜃))    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜏))
Theorem	bj-sylget 36609	Dual statement of sylgt 1822. Closed form of bj-sylge 36612. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜒 → 𝜑) → ((∃𝑥𝜑 → 𝜓) → (∃𝑥𝜒 → 𝜓)))
Theorem	bj-sylget2 36610	Uncurried (imported) form of bj-sylget 36609. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜓 → 𝜒)) → (∃𝑥𝜑 → 𝜒))
Theorem	bj-exlimg 36611	The general form of the *exlim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in a consequent, one can deduce from the universally quantified implication an implication where the antecedent is existentially quantified. Dual of bj-alrimg 36607. (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → (∃𝑥𝜒 → 𝜓)))
Theorem	bj-sylge 36612	Dual statement of sylg 1823 (the final "e" in the label stands for "existential (version of sylg 1823)". Variant of exlimih 2289. (Contributed by BJ, 25-Dec-2023.)
⊢ (∃𝑥𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimd 36613	A slightly more general exlimd 2219. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2219. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (∃𝑥𝜃 → 𝜏))    &   ⊢ (𝜓 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∃𝑥𝜒 → 𝜏))
Theorem	bj-nfimexal 36614	A weak from of nonfreeness in either an antecedent or a consequent implies that a universally quantified implication is equivalent to the associated implication where the antecedent is existentially quantified and the consequent is universally quantified. The forward implication always holds (this is 19.38 1839) and the converse implication is the join of instances of bj-alrimg 36607 and bj-exlimg 36611 (see 19.38a 1840 and 19.38b 1841). TODO: prove a version where the antecedents use the nonfreeness quantifier. (Contributed by BJ, 9-Dec-2023.)
⊢ (((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-alexim 36615	Closed form of aleximi 1832. Note: this proof is shorter, so aleximi 1832 could be deduced from it (exim 1834 would have to be proved first, see bj-eximALT 36629 but its proof is shorter (currently almost a subproof of aleximi 1832)). (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))
Theorem	bj-nexdh 36616	Closed form of nexdh 1865 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdh2 36617	Uncurried (imported) form of bj-nexdh 36616. (Contributed by BJ, 6-May-2019.)
⊢ ((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))
Theorem	bj-hbxfrbi 36618	Closed form of hbxfrbi 1825. Note: it is less important than nfbiit 1851. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 36729) in order not to require sp 2184 (modal T). See bj-hbyfrbi 36619 for its version with existential quantifiers. (Contributed by BJ, 6-May-2019.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
Theorem	bj-hbyfrbi 36619	Version of bj-hbxfrbi 36618 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓)))
Theorem	bj-exalim 36620	Distribute quantifiers over a nested implication. This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1910. I propose to move to the main part: bj-exalim 36620, bj-exalimi 36621, bj-exalims 36622, bj-exalimsi 36623, bj-ax12i 36625, bj-ax12wlem 36632, bj-ax12w 36665. A new label is needed for bj-ax12i 36625 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to ∃𝑥 in speimfw 1963 and spimfw 1965 (other spim* theorems use ∃𝑥 and very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))
Theorem	bj-exalimi 36621	An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 36620 (using mpg 1797) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1963 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))
Theorem	bj-exalims 36622	Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1965 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)))
Theorem	bj-exalimsi 36623	An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1965 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-ax12ig 36624	A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 36625. (Contributed by BJ, 19-Dec-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-ax12i 36625	A weakening of bj-ax12ig 36624 that is sufficient to prove a weak form of the axiom of substitution ax-12 2178. The general statement of which ax12i 1966 is an instance. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-nfimt 36626	Closed form of nfim 1896 and curried (exported) form of nfimt 1895. (Contributed by BJ, 20-Oct-2021.)
⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓)))
Theorem	bj-cbvalimt 36627	A lemma in closed form used to prove bj-cbval 36637 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1877 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.)
⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦(∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓)))))
Theorem	bj-cbveximt 36628	A lemma in closed form used to prove bj-cbvex 36638 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1877 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.)
⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))))
Theorem	bj-eximALT 36629	Alternate proof of exim 1834 directly from alim 1810 by using df-ex 1780 (using duality of ∀ and ∃. (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-aleximiALT 36630	Alternate proof of aleximi 1832 from exim 1834, which is sometimes used as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	bj-eximcom 36631	A commuted form of exim 1834 which is sometimes posited as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.)
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
21.19.4.3  Adding ax-5
Theorem	bj-ax12wlem 36632*	A lemma used to prove a weak version of the axiom of substitution ax-12 2178. (Temporary comment: The general statement that ax12wlem 2133 proves.) (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-cbvalim 36633*	A lemma used to prove bj-cbval 36637 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-cbvexim 36634*	A lemma used to prove bj-cbvex 36638 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜑 → ∃𝑦𝜓)))
Theorem	bj-cbvalimi 36635*	An equality-free general instance of one half of a precise form of bj-cbval 36637. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ ∀𝑦∃𝑥𝜒    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	bj-cbveximi 36636*	An equality-free general instance of one half of a precise form of bj-cbvex 36638. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ ∀𝑥∃𝑦𝜒    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbval 36637*	Changing a bound variable (universal quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦    &   ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	bj-cbvex 36638*	Changing a bound variable (existential quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦    &   ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Syntax	wmoo 36639	Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑
Definition	df-bj-mo 36640*	Definition of the uniqueness quantifier which is correct on the empty domain. Instead of the fresh variable 𝑧, one could save a dummy variable by using 𝑥 or 𝑦 at the cost of having nested quantifiers on the same variable. (Contributed by BJ, 12-Mar-2023.)
⊢ (∃**𝑥𝜑 ↔ ∀𝑧∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
21.19.4.4  Equality and substitution
Theorem	bj-ssbeq 36641*	Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 1967. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 36641 first with a DV condition on 𝑥, 𝑡, and then in the general case. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ ([𝑡 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧)
Theorem	bj-ssblem1 36642*	A lemma for the definiens of df-sb 2066. An instance of sp 2184 proved without it. Note: it has a common subproof with sbjust 2064. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-ssblem2 36643*	An instance of ax-11 2158 proved without it. The converse may not be provable without ax-11 2158 (since using alcomimw 2043 would require a DV on 𝜑, 𝑥, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
Theorem	bj-ax12v 36644*	A weaker form of ax-12 2178 and ax12v 2179, namely the generalization over 𝑥 of the latter. In this statement, all occurrences of 𝑥 are bound. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
Theorem	bj-ax12 36645*	Remove a DV condition from bj-ax12v 36644 (using core axioms only). (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
Theorem	bj-ax12ssb 36646*	Axiom bj-ax12 36645 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑)
Theorem	bj-19.41al 36647	Special case of 19.41 2236 proved from core axioms, ax-10 2142 (modal5), and hba1 2293 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	bj-equsexval 36648*	Special case of equsexv 2269 proved from core axioms, ax-10 2142 (modal5), and hba1 2293 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓)
Theorem	bj-subst 36649*	Proof of sbalex 2243 from core axioms, ax-10 2142 (modal5), and bj-ax12 36645. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	bj-ssbid2 36650	A special case of sbequ2 2250. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid2ALT 36651	Alternate proof of bj-ssbid2 36650, not using sbequ2 2250. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid1 36652	A special case of sbequ1 2249. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ssbid1ALT 36653	Alternate proof of bj-ssbid1 36652, not using sbequ1 2249. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ax6elem1 36654*	Lemma for bj-ax6e 36656. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	bj-ax6elem2 36655*	Lemma for bj-ax6e 36656. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Theorem	bj-ax6e 36656	Proof of ax6e 2381 (hence ax6 2382) from Tarski's system, ax-c9 38883, ax-c16 38885. Remark: ax-6 1967 is used only via its principal (unbundled) instance ax6v 1968. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦
21.19.4.5  Adding ax-6
Theorem	bj-spimvwt 36657*	Closed form of spimvw 1986. See also spimt 2384. (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-spnfw 36658	Theorem close to a closed form of spnfw 1979. (Contributed by BJ, 12-May-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbvexiw 36659*	Change bound variable. This is to cbvexvw 2037 what cbvaliw 2006 is to cbvalvw 2036. TODO: move after cbvalivw 2007. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbvexivw 36660*	Change bound variable. This is to cbvexvw 2037 what cbvalivw 2007 is to cbvalvw 2036. TODO: move after cbvalivw 2007. (Contributed by BJ, 17-Mar-2020.)
⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-modald 36661	A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	bj-denot 36662*	A weakening of ax-6 1967 and ax6v 1968. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
⊢ (𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)
Theorem	bj-eqs 36663*	A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2370. The disjoint variable condition DV (𝑥, 𝜑) is necessary for both directions: consider substituting 𝑥 = 𝑧 for 𝜑. (Contributed by BJ, 25-May-2021.)
⊢ (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
21.19.4.6  Adding ax-7
Theorem	bj-cbvexw 36664*	Change bound variable. This is to cbvexvw 2037 what cbvalw 2035 is to cbvalvw 2036. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Theorem	bj-ax12w 36665*	The general statement that ax12w 2134 proves. (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓)))
21.19.4.7  Membership predicate, ax-8 and ax-9
Theorem	bj-ax89 36666	A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2111 and ax-9 2119. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2111 and ax-9 2119, as proved here. In the other direction, one can prove ax-8 2111 (respectively ax-9 2119) from bj-ax89 36666 by using mpan2 691 (respectively mpan 690) and equid 2012. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))
Theorem	bj-cleljusti 36667*	One direction of cleljust 2118, requiring only ax-1 6-- ax-5 1910 and ax8v1 2113. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
21.19.4.8  Adding ax-11
Theorem	bj-alcomexcom 36668	Commutation of two existential quantifiers on a formula is equivalent to commutation of two universal quantifiers over the same variables on the negation of that formula. Can be placed in the ax-4 1809 section, soon after 2nexaln 1830, and used to prove excom 2163. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))
Theorem	bj-hbalt 36669	Closed form of hbal 2168. When in main part, prove hbal 2168 and hbald 2169 from it. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
21.19.4.9  Adding ax-12
Theorem	axc11n11 36670	Proof of axc11n 2424 from { ax-1 6-- ax-7 2008, axc11 2428 } . Almost identical to axc11nfromc11 38919. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	axc11n11r 36671	Proof of axc11n 2424 from { ax-1 6-- ax-7 2008, axc9 2380, axc11r 2366 } (note that axc16 2262 is provable from { ax-1 6-- ax-7 2008, axc11r 2366 }). Note that axc11n 2424 proves (over minimal calculus) that axc11 2428 and axc11r 2366 are equivalent. Therefore, axc11n11 36670 and axc11n11r 36671 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2428 appears slightly stronger since axc11n11r 36671 requires axc9 2380 while axc11n11 36670 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	bj-axc16g16 36672*	Proof of axc16g 2261 from { ax-1 6-- ax-7 2008, axc16 2262 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	bj-ax12v3 36673*	A weak version of ax-12 2178 which is stronger than ax12v 2179. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2012), then bj-ax12v3 36673 implies ax-5 1910 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 36674. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-ax12v3ALT 36674*	Alternate proof of bj-ax12v3 36673. Uses axc11r 2366 and axc15 2420 instead of ax-12 2178. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-sb 36675*	A weak variant of sbid2 2506 not requiring ax-13 2370 nor ax-10 2142. On top of Tarski's FOL, one implication requires only ax12v 2179, and the other requires only sp 2184. (Contributed by BJ, 25-May-2021.)
⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-modalbe 36676	The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2318. (Contributed by BJ, 20-Oct-2019.)
⊢ (𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	bj-spst 36677	Closed form of sps 2186. Once in main part, prove sps 2186 and spsd 2188 from it. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-19.21bit 36678	Closed form of 19.21bi 2190. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
Theorem	bj-19.23bit 36679	Closed form of 19.23bi 2192. (Contributed by BJ, 20-Oct-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓))
Theorem	bj-nexrt 36680	Closed form of nexr 2193. Contrapositive of 19.8a 2182. (Contributed by BJ, 20-Oct-2019.)
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
Theorem	bj-alrim 36681	Closed form of alrimi 2214. (Contributed by BJ, 2-May-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-alrim2 36682	Uncurried (imported) form of bj-alrim 36681. (Contributed by BJ, 2-May-2019.)
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓))
Theorem	bj-nfdt0 36683	A theorem close to a closed form of nf5d 2284 and nf5dh 2148. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))
Theorem	bj-nfdt 36684	Closed form of nf5d 2284 and nf5dh 2148. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
Theorem	bj-nexdt 36685	Closed form of nexd 2222. (Contributed by BJ, 20-Oct-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdvt 36686*	Closed form of nexdv 1936. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
Theorem	bj-alexbiex 36687	Adding a second quantifier over the same variable is a transparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-exexbiex 36688	Adding a second quantifier over the same variable is a transparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∃𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-alalbial 36689	Adding a second quantifier over the same variable is a transparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	bj-exalbial 36690	Adding a second quantifier over the same variable is a transparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∃𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	bj-19.9htbi 36691	Strengthening 19.9ht 2319 by replacing its consequent with a biconditional (19.9t 2205 does have a biconditional consequent). This propagates. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))
Theorem	bj-hbntbi 36692	Strengthening hbnt 2294 by replacing its consequent with a biconditional. See also hbntg 35793 and hbntal 44543. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 36691. (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
Theorem	bj-biexal1 36693	A general FOL biconditional that generalizes 19.9ht 2319 among others. For this and the following theorems, see also 19.35 1877, 19.21 2208, 19.23 2212. When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexal2 36694	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(∃𝑥𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexal3 36695	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑 → 𝜓))
Theorem	bj-bialal 36696	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexex 36697	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-hbext 36698	Closed form of hbex 2324. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))
Theorem	bj-nfalt 36699	Closed form of nfal 2322. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)
Theorem	bj-nfext 36700	Closed form of nfex 2323. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)