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-alrimg 36601	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 36605. (Contributed by BJ, 9-Dec-2023.)
⊢ ((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓 → 𝜒) → (𝜑 → ∀𝑥𝜒)))
Theorem	bj-alrimd 36602	A slightly more general alrimd 2212. A common usage will have 𝜑 substituted for 𝜓 and 𝜒 substituted for 𝜃, giving a form closer to alrimd 2212. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜃))    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜏))
Theorem	bj-sylget 36603	Dual statement of sylgt 1818. Closed form of bj-sylge 36606. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜒 → 𝜑) → ((∃𝑥𝜑 → 𝜓) → (∃𝑥𝜒 → 𝜓)))
Theorem	bj-sylget2 36604	Uncurried (imported) form of bj-sylget 36603. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜓 → 𝜒)) → (∃𝑥𝜑 → 𝜒))
Theorem	bj-exlimg 36605	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 36601. (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → (∃𝑥𝜒 → 𝜓)))
Theorem	bj-sylge 36606	Dual statement of sylg 1819 (the final "e" in the label stands for "existential (version of sylg 1819)". Variant of exlimih 2287. (Contributed by BJ, 25-Dec-2023.)
⊢ (∃𝑥𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimd 36607	A slightly more general exlimd 2215. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2215. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (∃𝑥𝜃 → 𝜏))    &   ⊢ (𝜓 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∃𝑥𝜒 → 𝜏))
Theorem	bj-nfimexal 36608	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 1835) and the converse implication is the join of instances of bj-alrimg 36601 and bj-exlimg 36605 (see 19.38a 1836 and 19.38b 1837). 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 36609	Closed form of aleximi 1828. Note: this proof is shorter, so aleximi 1828 could be deduced from it (exim 1830 would have to be proved first, see bj-eximALT 36623 but its proof is shorter (currently almost a subproof of aleximi 1828)). (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))
Theorem	bj-nexdh 36610	Closed form of nexdh 1862 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdh2 36611	Uncurried (imported) form of bj-nexdh 36610. (Contributed by BJ, 6-May-2019.)
⊢ ((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))
Theorem	bj-hbxfrbi 36612	Closed form of hbxfrbi 1821. Note: it is less important than nfbiit 1847. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 36723) in order not to require sp 2180 (modal T). See bj-hbyfrbi 36613 for its version with existential quantifiers. (Contributed by BJ, 6-May-2019.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
Theorem	bj-hbyfrbi 36613	Version of bj-hbxfrbi 36612 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓)))
Theorem	bj-exalim 36614	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 1907. I propose to move to the main part: bj-exalim 36614, bj-exalimi 36615, bj-exalims 36616, bj-exalimsi 36617, bj-ax12i 36619, bj-ax12wlem 36626, bj-ax12w 36659. A new label is needed for bj-ax12i 36619 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to ∃𝑥 in speimfw 1960 and spimfw 1962 (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 36615	An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 36614 (using mpg 1793) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1960 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))
Theorem	bj-exalims 36616	Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1962 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)))
Theorem	bj-exalimsi 36617	An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1962 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-ax12ig 36618	A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 36619. (Contributed by BJ, 19-Dec-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-ax12i 36619	A weakening of bj-ax12ig 36618 that is sufficient to prove a weak form of the axiom of substitution ax-12 2174. The general statement of which ax12i 1963 is an instance. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-nfimt 36620	Closed form of nfim 1893 and curried (exported) form of nfimt 1892. (Contributed by BJ, 20-Oct-2021.)
⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓)))
Theorem	bj-cbvalimt 36621	A lemma in closed form used to prove bj-cbval 36631 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1874 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 36622	A lemma in closed form used to prove bj-cbvex 36632 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1874 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 36623	Alternate proof of exim 1830 directly from alim 1806 by using df-ex 1776 (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 36624	Alternate proof of aleximi 1828 from exim 1830, 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 36625	A commuted form of exim 1830 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 36626*	A lemma used to prove a weak version of the axiom of substitution ax-12 2174. (Temporary comment: The general statement that ax12wlem 2129 proves.) (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-cbvalim 36627*	A lemma used to prove bj-cbval 36631 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 36628*	A lemma used to prove bj-cbvex 36632 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 36629*	An equality-free general instance of one half of a precise form of bj-cbval 36631. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ ∀𝑦∃𝑥𝜒    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	bj-cbveximi 36630*	An equality-free general instance of one half of a precise form of bj-cbvex 36632. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ ∀𝑥∃𝑦𝜒    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbval 36631*	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 36632*	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 36633	Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑
Definition	df-bj-mo 36634*	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 36635*	Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 1964. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 36635 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 36636*	A lemma for the definiens of df-sb 2062. An instance of sp 2180 proved without it. Note: it has a common subproof with sbjust 2060. (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 36637*	An instance of ax-11 2154 proved without it. The converse may not be provable without ax-11 2154 (since using alcomimw 2039 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 36638*	A weaker form of ax-12 2174 and ax12v 2175, 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 36639*	Remove a DV condition from bj-ax12v 36638 (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 36640*	Axiom bj-ax12 36639 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑)
Theorem	bj-19.41al 36641	Special case of 19.41 2232 proved from core axioms, ax-10 2138 (modal5), and hba1 2291 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	bj-equsexval 36642*	Special case of equsexv 2265 proved from core axioms, ax-10 2138 (modal5), and hba1 2291 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓)
Theorem	bj-subst 36643*	Proof of sbalex 2239 from core axioms, ax-10 2138 (modal5), and bj-ax12 36639. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	bj-ssbid2 36644	A special case of sbequ2 2246. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid2ALT 36645	Alternate proof of bj-ssbid2 36644, not using sbequ2 2246. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid1 36646	A special case of sbequ1 2245. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ssbid1ALT 36647	Alternate proof of bj-ssbid1 36646, not using sbequ1 2245. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ax6elem1 36648*	Lemma for bj-ax6e 36650. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	bj-ax6elem2 36649*	Lemma for bj-ax6e 36650. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Theorem	bj-ax6e 36650	Proof of ax6e 2385 (hence ax6 2386) from Tarski's system, ax-c9 38871, ax-c16 38873. Remark: ax-6 1964 is used only via its principal (unbundled) instance ax6v 1965. (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 36651*	Closed form of spimvw 1992. See also spimt 2388. (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-spnfw 36652	Theorem close to a closed form of spnfw 1976. (Contributed by BJ, 12-May-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbvexiw 36653*	Change bound variable. This is to cbvexvw 2033 what cbvaliw 2002 is to cbvalvw 2032. TODO: move after cbvalivw 2003. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbvexivw 36654*	Change bound variable. This is to cbvexvw 2033 what cbvalivw 2003 is to cbvalvw 2032. TODO: move after cbvalivw 2003. (Contributed by BJ, 17-Mar-2020.)
⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-modald 36655	A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	bj-denot 36656*	A weakening of ax-6 1964 and ax6v 1965. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
⊢ (𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)
Theorem	bj-eqs 36657*	A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2374. 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 36658*	Change bound variable. This is to cbvexvw 2033 what cbvalw 2031 is to cbvalvw 2032. (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 36659*	The general statement that ax12w 2130 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 36660	A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2107 and ax-9 2115. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2107 and ax-9 2115, as proved here. In the other direction, one can prove ax-8 2107 (respectively ax-9 2115) from bj-ax89 36660 by using mpan2 691 (respectively mpan 690) and equid 2008. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))
Theorem	bj-cleljusti 36661*	One direction of cleljust 2114, requiring only ax-1 6-- ax-5 1907 and ax8v1 2109. (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 36662	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 1805 section, soon after 2nexaln 1826, and used to prove excom 2159. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))
Theorem	bj-hbalt 36663	Closed form of hbal 2164. When in main part, prove hbal 2164 and hbald 2165 from it. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
21.19.4.9  Adding ax-12
Theorem	axc11n11 36664	Proof of axc11n 2428 from { ax-1 6-- ax-7 2004, axc11 2432 } . Almost identical to axc11nfromc11 38907. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	axc11n11r 36665	Proof of axc11n 2428 from { ax-1 6-- ax-7 2004, axc9 2384, axc11r 2368 } (note that axc16 2258 is provable from { ax-1 6-- ax-7 2004, axc11r 2368 }). Note that axc11n 2428 proves (over minimal calculus) that axc11 2432 and axc11r 2368 are equivalent. Therefore, axc11n11 36664 and axc11n11r 36665 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2432 appears slightly stronger since axc11n11r 36665 requires axc9 2384 while axc11n11 36664 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	bj-axc16g16 36666*	Proof of axc16g 2257 from { ax-1 6-- ax-7 2004, axc16 2258 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	bj-ax12v3 36667*	A weak version of ax-12 2174 which is stronger than ax12v 2175. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2008), then bj-ax12v3 36667 implies ax-5 1907 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 36668. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-ax12v3ALT 36668*	Alternate proof of bj-ax12v3 36667. Uses axc11r 2368 and axc15 2424 instead of ax-12 2174. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-sb 36669*	A weak variant of sbid2 2510 not requiring ax-13 2374 nor ax-10 2138. On top of Tarski's FOL, one implication requires only ax12v 2175, and the other requires only sp 2180. (Contributed by BJ, 25-May-2021.)
⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-modalbe 36670	The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2317. (Contributed by BJ, 20-Oct-2019.)
⊢ (𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	bj-spst 36671	Closed form of sps 2182. Once in main part, prove sps 2182 and spsd 2184 from it. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-19.21bit 36672	Closed form of 19.21bi 2186. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
Theorem	bj-19.23bit 36673	Closed form of 19.23bi 2188. (Contributed by BJ, 20-Oct-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓))
Theorem	bj-nexrt 36674	Closed form of nexr 2189. Contrapositive of 19.8a 2178. (Contributed by BJ, 20-Oct-2019.)
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
Theorem	bj-alrim 36675	Closed form of alrimi 2210. (Contributed by BJ, 2-May-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-alrim2 36676	Uncurried (imported) form of bj-alrim 36675. (Contributed by BJ, 2-May-2019.)
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓))
Theorem	bj-nfdt0 36677	A theorem close to a closed form of nf5d 2282 and nf5dh 2144. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))
Theorem	bj-nfdt 36678	Closed form of nf5d 2282 and nf5dh 2144. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
Theorem	bj-nexdt 36679	Closed form of nexd 2218. (Contributed by BJ, 20-Oct-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdvt 36680*	Closed form of nexdv 1933. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
Theorem	bj-alexbiex 36681	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 36682	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 36683	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 36684	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 36685	Strengthening 19.9ht 2318 by replacing its consequent with a biconditional (19.9t 2201 does have a biconditional consequent). This propagates. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))
Theorem	bj-hbntbi 36686	Strengthening hbnt 2292 by replacing its consequent with a biconditional. See also hbntg 35786 and hbntal 44550. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 36685. (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
Theorem	bj-biexal1 36687	A general FOL biconditional that generalizes 19.9ht 2318 among others. For this and the following theorems, see also 19.35 1874, 19.21 2204, 19.23 2208. 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 36688	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 36689	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 36690	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 36691	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 36692	Closed form of hbex 2323. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))
Theorem	bj-nfalt 36693	Closed form of nfal 2321. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)
Theorem	bj-nfext 36694	Closed form of nfex 2322. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)
Theorem	bj-eeanvw 36695*	Version of exdistrv 1952 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2154. (The same can be done with eeeanv 2350 and ee4anv 2351.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	bj-modal4 36696	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 36697. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	bj-modal4e 36697	First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 36696 (hba1 2291). (Contributed by BJ, 21-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)
Theorem	bj-modalb 36698	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 36699	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 36700	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))