P. 378 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	bj-spimnfe 37701	A universal specification result: if 𝜑 is true for all values of 𝑥 and implies 𝜓 for at least one value, and if furthermore 𝑥 is ∃-weakly nonfree in 𝜓, then 𝜓 follows. An intermediate result on the way to prove 19.36i 2271, bj-19.36im 37843, 19.36imv 1980, spimfw 2000... (Contributed by BJ, 3-Apr-2026.) Proof should not use 19.35 1912. (Proof modification is discouraged.)
⊢ ((∃𝑥𝜓 → 𝜓) → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-spimenfa 37702	An existential generalization result: if 𝜑 holds and implies 𝜓 for at least one value of 𝑥, and if furthermore 𝑥 is ∀ -weakly nonfree in 𝜑, then 𝜓 holds for at least one value of 𝑥. (Contributed by BJ, 3-Apr-2026.) Proof should not use 19.35 1912. (Proof modification is discouraged.)
⊢ ((𝜑 → ∀𝑥𝜑) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))
Theorem	bj-spim 37703	A lemma for universal specification. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 2004 will prove Hypothesis bj-spim.denote. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))    &   ⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑥𝜒 → 𝜃))
Theorem	bj-spime 37704	A lemma for existential generalization. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 2004 will prove Hypothesis bj-spime.denote. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 → ∃𝑥𝜃))
Theorem	bj-cbvalimd0 37705	A lemma for alpha-renaming of variables bound by a universal quantifier. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 2004 will prove Hypothesis bj-cbvalimd0.denote. When ax6ev 2004 is not available but only its universal closure is, then bj-cbvalimd 37708 or bj-cbvalimdv 37710 should be used (see bj-cbvalimdlem 37706, bj-cbval 37723). (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))    &   ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))    &   ⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))
Theorem	bj-cbvalimdlem 37706	A lemma for alpha-renaming of variables bound by a universal quantifier. Hypothesis bj-cbvalimdlem.nfch can be proved either from DV conditions as in bj-cbvalimdv 37710 or from a nonfreeness condition and alcom 2199 as in bj-cbvalimd 37708. Hypothesis bj-cbvalimdlem.denote is weaker than the corresponding hypothesis of bj-cbvalimd0 37705, and this proof is therefore a bit longer, not using bj-spim 37703 but bj-eximcom 37694. (Contributed by BJ, 12-Mar-2023.) Proof should not use 19.35 1912. (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦∀𝑥𝜒))    &   ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))    &   ⊢ (𝜑 → ∀𝑦∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))
Theorem	bj-cbveximdlem 37707	A lemma for alpha-renaming of variables bound by an existential quantifier. Hypothesis bj-cbveximdlem.nfth can be proved either from DV conditions as in bj-cbveximdv 37711 or from a nonfreeness condition and excom 2202 as in bj-cbveximd 37709. Hypothesis bj-cbveximdlem.denote is weaker than the corresponding hypothesis of ~ bj-cbveximd0 , and this proof is therefore a bit longer, not using bj-spime 37704 but bj-eximcom 37694. (Contributed by BJ, 12-Mar-2023.) Proof should not use 19.35 1912. (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))    &   ⊢ (𝜑 → (∃𝑥∃𝑦𝜃 → ∃𝑦𝜃))    &   ⊢ (𝜑 → ∀𝑥∃𝑦𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∃𝑥𝜒 → ∃𝑦𝜃))
Theorem	bj-cbvalimd 37708	A lemma for alpha-renaming of variables bound by a universal quantifier. (Contributed by BJ, 4-Apr-2026.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))    &   ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))    &   ⊢ (𝜑 → ∀𝑦∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))
Theorem	bj-cbveximd 37709	A lemma for alpha-renaming of variables bound by an existential quantifier. (Contributed by BJ, 4-Apr-2026.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))    &   ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))    &   ⊢ (𝜑 → ∀𝑥∃𝑦𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∃𝑥𝜒 → ∃𝑦𝜃))
Theorem	bj-cbvalimdv 37710*	A lemma for alpha-renaming of variables bound by a universal quantifier. (Contributed by BJ, 4-Apr-2026.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))    &   ⊢ (𝜑 → ∀𝑦∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))
Theorem	bj-cbveximdv 37711*	A lemma for alpha-renaming of variables bound by an existential quantifier. (Contributed by BJ, 4-Apr-2026.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))    &   ⊢ (𝜑 → ∀𝑥∃𝑦𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∃𝑥𝜒 → ∃𝑦𝜃))
21.20.4.4  Adding ax-5
Theorem	bj-spvw 37712*	Version of spvw 2016 and 19.3v 2017 proved from ax-1 6-- ax-5 1945. The antecedent can for instance be proved with the existence axiom extru 2010. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (𝜓 ↔ ∀𝑥𝜓))
Theorem	bj-spvew 37713*	Version of 19.8v 2018 and 19.9v 2019 proved from ax-1 6-- ax-5 1945. The antecedent can for instance be proved with the existence axiom extru 2010. (Contributed by BJ, 8-Mar-2026.) This could also be proved from bj-spvw 37712 using duality, but that proof would not be intuitionistic, contrary to the present one. (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (𝜓 ↔ ∃𝑥𝜓))
Theorem	bj-alextruim 37714*	An equivalent expression for universal quantification over a non-occurring variable proved over ax-1 6-- ax-5 1945. The forward implication can be strengthened when ax-6 2002 is posited (which implies that models are non-empty), see spvw 2016. The reverse implication can be seen as a strengthening of ax-5 1945 (since the antecedent of the implication is weakened). See bj-exextruan 37715 for a dual statement. An approximate meaning is: the universal quantification of a proposition over a non-occurring variable holds if and only if the proposition holds in nonempty universes. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 ↔ (∃𝑥⊤ → 𝜑))
Theorem	bj-exextruan 37715*	An equivalent expression for existential quantification over a non-occurring variable proved over ax-1 6-- ax-5 1945. The forward implication can be seen as a strengthening of ax-5 1945 (a conjunct is added to the consequent of the implication). The reverse implication can be strengthened when ax-6 2002 is posited (which implies that models are non-empty), see 19.8v 2018. See bj-alextruim 37714 for a dual statement. An approximate meaning is: the existential quantification of a proposition over a non-occurring variable holds if and only if the proposition holds and the universe is nonempty. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 ↔ (∃𝑥⊤ ∧ 𝜑))
Theorem	bj-cbvalvv 37716*	Universally quantifying over a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1945 and the existence axiom extru 2010. See bj-cbvaw 37718 for a strengthening. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∀𝑦𝜓))
Theorem	bj-cbvexvv 37717*	Existentially quantifying over a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1945 and the existence axiom extru 2010. See bj-cbvew 37719 for a strengthening. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (∃𝑦𝜓 → ∃𝑥𝜓))
Theorem	bj-cbvaw 37718*	Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 37716. If ⊥ is substituted for 𝜑, then the statement reads: "universally quantifying over a non-occurring variable is independent from the variable as soon as that result is true for the False truth constant". The label "cbvaw" means "'change bound variable' theorem, 'all' quantifier, weak version". (Contributed by BJ, 14-Mar-2026.) This proof is not intuitionistic (it uses ja 188); an intuitionistically valid statement is obtained by expressing the antecedent as a disjunction (classically equivalent through imor 867). (Proof modification is discouraged.)
⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∀𝑥𝜓 → ∀𝑦𝜓))
Theorem	bj-cbvew 37719*	Existentially quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvexvv 37717. If ⊤ is substituted for 𝜑, then the statement reads: "existentially quantifying over a non-occurring variable is independent from the variable as soon as that result is true for the True truth constant. The label "cbvew" means "'change bound variable' theorem, 'exists' quantifier, weak version". (Contributed by BJ, 14-Mar-2026.) This proof is intuitionistic. (Proof modification is discouraged.)
⊢ ((∃𝑥⊤ → ∃𝑦𝜑) → (∃𝑥𝜓 → ∃𝑦𝜓))
Theorem	bj-cbveaw 37720*	Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 37716. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ((∃𝑥⊤ → ∃𝑦𝜑) → (∀𝑦𝜓 → ∀𝑥𝜓))
Theorem	bj-cbvaew 37721*	Exixtentially quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvexvv 37717. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∃𝑦𝜓 → ∃𝑥𝜓))
Theorem	bj-ax12wlem 37722*	A lemma used to prove a weak version of the axiom of substitution ax-12 2216. (Temporary comment: The general statement that ax12wlem 2171 proves.) (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-cbval 37723*	Changing a bound variable (universal quantification case) in a weak axiomatization that assumes that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) Proved from ax-1 6-- ax-5 1945. (Proof modification is discouraged.)
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦    &   ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥    &   ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)    &   ⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvex 37724*	Changing a bound variable (existential quantification case) in a weak axiomatization that assumes that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) Proved from ax-1 6-- ax-5 1945. (Proof modification is discouraged.)
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦    &   ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥    &   ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)    &   ⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Syntax	wmoo 37725	Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑
Definition	df-bj-mo 37726*	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))
21.20.4.5  Equality and substitution
Theorem	bj-df-sb 37727*	Proposed definition to replace df-sb 2102 and df-sbc 3743. Proof is therefore unimportant. Contrary to df-sb 2102, this definition makes a substituted formula false when one substitutes a non-existent object for a variable. This is better suited to the "Levy-style" treatment of classes as virtual objects adopted by set.mm. Indeed, in most set theories, and in particular in this database, one wants to be able to collect the extension of a formula, so it is natural that the extension of a formula contain only sets, and not proper classes. This is illustrated by the fact that bj-dfsbc 37729 holds unconditionally. That statement could therefore replace df-clab 2742. This definition is equivalent to df-sb 2102 when a setvar is substituted for the class, since, as soon as, ax6ev 2004 is posited, all variables "exist". (Contributed by BJ, 19-Feb-2026.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-sbcex 37728	Proof of sbcex 3752 when taking bj-df-sb 37727 as definition. (Contributed by BJ, 19-Feb-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
Theorem	bj-dfsbc 37729	Proof of df-sbc 3743 when taking bj-df-sb 37727 as definition. (Contributed by BJ, 19-Feb-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
Theorem	bj-ssbeq 37730*	Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 2002. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 37730 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 37731*	A lemma for the definiens of df-sb 2102. An instance of sp 2223 proved without it. Note: it has a common subproof with sbjust 2100. (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 37732*	An instance of ax-11 2197 proved without it. The converse may not be provable without ax-11 2197 (since using alcomimw 2078 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 37733*	A weaker form of ax-12 2216 and ax12v 2217, 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 37734*	Remove a DV condition from bj-ax12v 37733 (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 37735*	Axiom bj-ax12 37734 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑)
Theorem	bj-19.41al 37736	Special case of 19.41 2275 proved from core axioms, ax-10 2180 (modal5), and hba1 2330 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	bj-equsexval 37737*	Special case of equsexv 2306 proved from core axioms, ax-10 2180 (modal5), and hba1 2330 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓)
Theorem	bj-subst 37738*	Proof of sbalex 2282 from core axioms, ax-10 2180 (modal5), and bj-ax12 37734. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	bj-ssbid2 37739	A special case of sbequ2 2288. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid2ALT 37740	Alternate proof of bj-ssbid2 37739, not using sbequ2 2288. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid1 37741	A special case of sbequ1 2287. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ssbid1ALT 37742	Alternate proof of bj-ssbid1 37741, not using sbequ1 2287. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ax6elem1 37743*	Lemma for bj-ax6e 37745. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	bj-ax6elem2 37744*	Lemma for bj-ax6e 37745. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Theorem	bj-ax6e 37745	Proof of ax6e 2415 (hence ax6 2416) from Tarski's system, ax-c9 40139, ax-c16 40141. Remark: ax-6 2002 is used only via its principal (unbundled) instance ax6v 2003. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦
21.20.4.6  Adding ax-6
Theorem	bj-spim0 37746*	A universal specialization result in deduction form, proved from ax-1 6 -- ax-6 2002, 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 → (∃𝑥𝜒 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-spimvwt 37747*	Closed form of spimvw 2021. See also spimt 2418. (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-spnfw 37748	Theorem close to a closed form of spnfw 2014. (Contributed by BJ, 12-May-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbvexiw 37749*	Change bound variable. This is to cbvexvw 2072 what cbvaliw 2041 is to cbvalvw 2071. TODO: move after cbvalivw 2042. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbvexivw 37750*	Change bound variable. This is to cbvexvw 2072 what cbvalivw 2042 is to cbvalvw 2071. TODO: move after cbvalivw 2042. (Contributed by BJ, 17-Mar-2020.)
⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-modald 37751	A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	bj-denot 37752*	A weakening of ax-6 2002 and ax6v 2003. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
⊢ (𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)
Theorem	bj-eqs 37753*	A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2404. 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))
21.20.4.7  Adding ax-7
Theorem	bj-cbvexw 37754*	Change bound variable. This is to cbvexvw 2072 what cbvalw 2070 is to cbvalvw 2071. (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 37755*	The general statement that ax12w 2172 proves. (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓)))
21.20.4.8  Membership predicate, ax-8 and ax-9
Theorem	bj-ax89 37756	A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2149 and ax-9 2157. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2149 and ax-9 2157, as proved here. In the other direction, one can prove ax-8 2149 (respectively ax-9 2157) from bj-ax89 37756 by using mpan2 704 (respectively mpan 703) and equid 2047. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))
Theorem	bj-cleljusti 37757*	One direction of cleljust 2156, requiring only ax-1 6-- ax-5 1945 and ax8v1 2151. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
21.20.4.9  Adding ax-11
Theorem	bj-alcomexcom 37758	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 1842 section, soon after 2nexaln 1863, and used to prove excom 2202. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))
Theorem	bj-hbald 37759	General statement that hbald 2208 proves . (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑦𝜓)    &   ⊢ (𝜓 → (𝜒 → ∀𝑥𝜃))    ⇒   ⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑥∀𝑦𝜃))
Theorem	bj-hbalt 37760	Closed form of (general instance of) hbal 2207. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑦(𝜑 → ∀𝑥𝜓) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-hbal 37761	More general instance of hbal 2207. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)
21.20.4.10  Adding ax-12
Theorem	axc11n11 37762	Proof of axc11n 2458 from { ax-1 6-- ax-7 2043, axc11 2462 } . Almost identical to axc11nfromc11 40175. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	axc11n11r 37763	Proof of axc11n 2458 from { ax-1 6-- ax-7 2043, axc9 2414, axc11r 2400 } (note that axc16 2299 is provable from { ax-1 6-- ax-7 2043, axc11r 2400 }). Theorem axc11n 2458 proves (over minimal calculus) that axc11 2462 and axc11r 2400 are equivalent. Therefore, axc11n11 37762 and axc11n11r 37763 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2462 appears slightly stronger since axc11n11r 37763 requires axc9 2414 while axc11n11 37762 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	bj-axc16g16 37764*	Proof of axc16g 2298 from { ax-1 6-- ax-7 2043, axc16 2299 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	bj-ax12v3 37765*	A weak version of ax-12 2216 which is stronger than ax12v 2217. If one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2047), then bj-ax12v3 37765 implies ax-5 1945 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 37766. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-ax12v3ALT 37766*	Alternate proof of bj-ax12v3 37765. Uses axc11r 2400 and axc15 2454 instead of ax-12 2216. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-sb 37767*	A weak variant of sbid2 2540 not requiring ax-13 2404 nor ax-10 2180. On top of Tarski's FOL, one implication requires only ax12v 2217, and the other requires only sp 2223. (Contributed by BJ, 25-May-2021.)
⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-modalbe 37768	The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2353. (Contributed by BJ, 20-Oct-2019.)
⊢ (𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	bj-spst 37769	Closed form of sps 2225. Once in main part, prove sps 2225 and spsd 2227 from it. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-19.21bit 37770	Closed form of 19.21bi 2229. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
Theorem	bj-19.23bit 37771	Closed form of 19.23bi 2231. (Contributed by BJ, 20-Oct-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓))
Theorem	bj-nexrt 37772	Closed form of nexr 2232. Contrapositive of 19.8a 2221. (Contributed by BJ, 20-Oct-2019.)
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
Theorem	bj-alrim 37773	Closed form of alrimi 2253. (Contributed by BJ, 2-May-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-alrim2 37774	Uncurried (imported) form of bj-alrim 37773. (Contributed by BJ, 2-May-2019.)
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓))
Theorem	bj-nfdt0 37775	A theorem close to a closed form of nf5d 2321 and nf5dh 2187. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))
Theorem	bj-nfdt 37776	Closed form of nf5d 2321 and nf5dh 2187. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
Theorem	bj-nexdt 37777	Closed form of nexd 2261. (Contributed by BJ, 20-Oct-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdvt 37778*	Closed form of nexdv 1971. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
Theorem	bj-alexbiex 37779	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 37780	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 37781	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 37782	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 37783	Strengthening 19.9ht 2354 by replacing its consequent with a biconditional (19.9t 2244 does have a biconditional consequent). This propagates. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))
Theorem	bj-hbntbi 37784	Strengthening hbnt 2331 by replacing its consequent with a biconditional. See also hbntg 36745 and hbntal 45727. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 37783. (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
Theorem	bj-biexal1 37785	A general FOL biconditional that generalizes 19.9ht 2354 among others. For this and the following theorems, see also 19.35 1912, 19.21 2247, 19.23 2251. 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 37786	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 37787	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 37788	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 37789	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-hbexd 37790	A more general instance of the deduction form of hbex 2359. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑦𝜓)    &   ⊢ (𝜓 → (𝜒 → ∀𝑥𝜃))    ⇒   ⊢ (𝜑 → (∃𝑦𝜒 → ∀𝑥∃𝑦𝜃))
Theorem	bj-hbext 37791	Closed form of bj-hbex 37792 and hbex 2359. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓))
Theorem	bj-hbex 37792	A more general instance of hbex 2359. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓)
Theorem	bj-nfalt 37793	Closed form of nfal 2357. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)
Theorem	bj-nfext 37794	Closed form of nfex 2358. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)
Theorem	bj-eeanvw 37795*	Version of exdistrv 1990 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2197. (The same can be done with eeeanv 2383 and ee4anv 2384.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	bj-modal4 37796	First-order logic form of the modal axiom (4). See hba1 2330. This is the standard proof of the implication in modal logic (B5 ⇒ 4). Its dual statement is bj-modal4e 37797. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	bj-modal4e 37797	First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 37796 (hba1 2330). (Contributed by BJ, 21-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)
Theorem	bj-modalb 37798	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 37799	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 37800	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))