P. 372 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37101-37200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-eximcom 37101	A commuted form of exim 1857 which is sometimes posited as an axiom in instuitionistic modal logic. Forward implication of 19.35 1900. Its converse is not intuitionistic. (Contributed by BJ, 9-Dec-2023.)
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-exalims 37102	Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1988 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)))
Theorem	bj-exalimsi 37103	An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1988 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-axdd2ALT 37104	Alternate proof of bj-axdd2 37047 (this should replace bj-axdd2 37047 when bj-exalimi 37100 is moved to the main section). (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
Theorem	bj-ax12ig 37105	A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 37106. (Contributed by BJ, 19-Dec-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-ax12i 37106	A weakening of bj-ax12ig 37105 that is sufficient to prove a weak form of the axiom of substitution ax-12 2215. The general statement of which ax12i 1989 is an instance. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-nfimt 37107	Closed form of nfim 1919 and curried (exported) form of nfimt 1918. (Contributed by BJ, 20-Oct-2021.) Proof should not use 19.35 1900. (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓)))
Theorem	bj-spimnfe 37108	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 2269, bj-19.36im 37250, 19.36imv 1968, spimfw 1988... (Contributed by BJ, 3-Apr-2026.) Proof should not use 19.35 1900. (Proof modification is discouraged.)
⊢ ((∃𝑥𝜓 → 𝜓) → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-spimenfa 37109	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 1900. (Proof modification is discouraged.)
⊢ ((𝜑 → ∀𝑥𝜑) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))
Theorem	bj-spim 37110	A lemma for universal specification. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1992 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 37111	A lemma for existential generalization. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1992 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 37112	A lemma for alpha-renaming of variables bound by a universal quantifier. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1992 will prove Hypothesis bj-cbvalimd0.denote. When ax6ev 1992 is not available but only its universal closure is, then bj-cbvalimd 37115 or bj-cbvalimdv 37117 should be used (see bj-cbvalimdlem 37113, bj-cbval 37130). (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 37113	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 37117 or from a nonfreeness condition and alcom 2196 as in bj-cbvalimd 37115. Hypothesis bj-cbvalimdlem.denote is weaker than the corresponding hypothesis of bj-cbvalimd0 37112, and this proof is therefore a bit longer, not using bj-spim 37110 but bj-eximcom 37101. (Contributed by BJ, 12-Mar-2023.) Proof should not use 19.35 1900. (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 37114	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 37118 or from a nonfreeness condition and excom 2199 as in bj-cbveximd 37116. 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 37111 but bj-eximcom 37101. (Contributed by BJ, 12-Mar-2023.) Proof should not use 19.35 1900. (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 37115	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 37116	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 37117*	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 37118*	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))
21.19.4.4  Adding ax-5
Theorem	bj-spvw 37119*	Version of spvw 2004 and 19.3v 2005 proved from ax-1 6-- ax-5 1933. The antecedent can for instance be proved with the existence axiom extru 1998. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (𝜓 ↔ ∀𝑥𝜓))
Theorem	bj-spvew 37120*	Version of 19.8v 2006 and 19.9v 2007 proved from ax-1 6-- ax-5 1933. The antecedent can for instance be proved with the existence axiom extru 1998. (Contributed by BJ, 8-Mar-2026.) This could also be proved from bj-spvw 37119 using duality, but that proof would not be intuitionistic, contrary to the present one. (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (𝜓 ↔ ∃𝑥𝜓))
Theorem	bj-alextruim 37121*	An equivalent expression for universal quantification over a non-occurring variable proved over ax-1 6-- ax-5 1933. The forward implication can be strengthened when ax-6 1990 is posited (which implies that models are non-empty), see spvw 2004. The reverse implication can be seen as a strengthening of ax-5 1933 (since the antecedent of the implication is weakened). See bj-exextruan 37122 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 37122*	An equivalent expression for existential quantification over a non-occurring variable proved over ax-1 6-- ax-5 1933. The forward implication can be seen as a strengthening of ax-5 1933 (a conjunct is added to the consequent of the implication). The reverse implication can be strengthened when ax-6 1990 is posited (which implies that models are non-empty), see 19.8v 2006. See bj-alextruim 37121 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 37123*	Universally quantifying over a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1933 and the existence axiom extru 1998. See bj-cbvaw 37125 for a strengthening. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∀𝑦𝜓))
Theorem	bj-cbvexvv 37124*	Existentially quantifying over a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1933 and the existence axiom extru 1998. See bj-cbvew 37126 for a strengthening. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (∃𝑦𝜓 → ∃𝑥𝜓))
Theorem	bj-cbvaw 37125*	Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 37123. 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 866). (Proof modification is discouraged.)
⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∀𝑥𝜓 → ∀𝑦𝜓))
Theorem	bj-cbvew 37126*	Existentially quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvexvv 37124. 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 37127*	Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 37123. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ((∃𝑥⊤ → ∃𝑦𝜑) → (∀𝑦𝜓 → ∀𝑥𝜓))
Theorem	bj-cbvaew 37128*	Exixtentially quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvexvv 37124. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∃𝑦𝜓 → ∃𝑥𝜓))
Theorem	bj-ax12wlem 37129*	A lemma used to prove a weak version of the axiom of substitution ax-12 2215. (Temporary comment: The general statement that ax12wlem 2169 proves.) (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-cbval 37130*	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 1933. (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 37131*	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 1933. (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 37132	Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑
Definition	df-bj-mo 37133*	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))
21.19.4.5  Equality and substitution
Theorem	bj-df-sb 37134*	Proposed definition to replace df-sb 2094 and df-sbc 3748. Proof is therefore unimportant. Contrary to df-sb 2094, 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. That difference is unimportant since as soon as ax6ev 1992 is posited, all variables "exist". (Contributed by BJ, 19-Feb-2026.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-sbcex 37135	Proof of sbcex 3757 when taking bj-df-sb 37134 as definition. (Contributed by BJ, 19-Feb-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
Theorem	bj-dfsbc 37136	Proof of df-sbc 3748 when taking bj-df-sb 37134 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 37137*	Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 1990. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 37137 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 37138*	A lemma for the definiens of df-sb 2094. An instance of sp 2221 proved without it. Note: it has a common subproof with rename-sb 2092. (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 37139*	An instance of ax-11 2194 proved without it. The converse may not be provable without ax-11 2194 (since using alcomimw 2066 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 37140*	A weaker form of ax-12 2215 and ax12v 2216, 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 37141*	Remove a DV condition from bj-ax12v 37140 (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 37142*	Axiom bj-ax12 37141 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑)
Theorem	bj-19.41al 37143	Special case of 19.41 2273 proved from core axioms, ax-10 2178 (modal5), and hba1 2330 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	bj-equsexval 37144*	Special case of equsexv 2306 proved from core axioms, ax-10 2178 (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 37145*	Proof of sbalex 2280 from core axioms, ax-10 2178 (modal5), and bj-ax12 37141. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	bj-ssbid2 37146	A special case of sbequ2 2287. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid2ALT 37147	Alternate proof of bj-ssbid2 37146, not using sbequ2 2287. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid1 37148	A special case of sbequ1 2286. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ssbid1ALT 37149	Alternate proof of bj-ssbid1 37148, not using sbequ1 2286. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ax6elem1 37150*	Lemma for bj-ax6e 37152. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	bj-ax6elem2 37151*	Lemma for bj-ax6e 37152. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Theorem	bj-ax6e 37152	Proof of ax6e 2417 (hence ax6 2418) from Tarski's system, ax-c9 39526, ax-c16 39528. Remark: ax-6 1990 is used only via its principal (unbundled) instance ax6v 1991. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦
21.19.4.6  Adding ax-6
Theorem	bj-spim0 37153*	A universal specialization result in deduction form, proved from ax-1 6 -- ax-6 1990, 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 37154*	Closed form of spimvw 2009. See also spimt 2420. (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-spnfw 37155	Theorem close to a closed form of spnfw 2002. (Contributed by BJ, 12-May-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbvexiw 37156*	Change bound variable. This is to cbvexvw 2060 what cbvaliw 2029 is to cbvalvw 2059. TODO: move after cbvalivw 2030. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbvexivw 37157*	Change bound variable. This is to cbvexvw 2060 what cbvalivw 2030 is to cbvalvw 2059. TODO: move after cbvalivw 2030. (Contributed by BJ, 17-Mar-2020.)
⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-modald 37158	A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	bj-denot 37159*	A weakening of ax-6 1990 and ax6v 1991. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
⊢ (𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)
Theorem	bj-eqs 37160*	A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2406. 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))
21.19.4.7  Adding ax-7
Theorem	bj-cbvexw 37161*	Change bound variable. This is to cbvexvw 2060 what cbvalw 2058 is to cbvalvw 2059. (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 37162*	The general statement that ax12w 2170 proves. (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓)))
21.19.4.8  Membership predicate, ax-8 and ax-9
Theorem	bj-ax89 37163	A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2147 and ax-9 2155. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2147 and ax-9 2155, as proved here. In the other direction, one can prove ax-8 2147 (respectively ax-9 2155) from bj-ax89 37163 by using mpan2 703 (respectively mpan 702) and equid 2035. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))
Theorem	bj-cleljusti 37164*	One direction of cleljust 2154, requiring only ax-1 6-- ax-5 1933 and ax8v1 2149. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
21.19.4.9  Adding ax-11
Theorem	bj-alcomexcom 37165	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 1832 section, soon after 2nexaln 1853, and used to prove excom 2199. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))
Theorem	bj-hbald 37166	General statement that hbald 2205 proves . (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑦𝜓)    &   ⊢ (𝜓 → (𝜒 → ∀𝑥𝜃))    ⇒   ⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑥∀𝑦𝜃))
Theorem	bj-hbalt 37167	Closed form of (general instance of) hbal 2204. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑦(𝜑 → ∀𝑥𝜓) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-hbal 37168	More general instance of hbal 2204. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)
21.19.4.10  Adding ax-12
Theorem	axc11n11 37169	Proof of axc11n 2460 from { ax-1 6-- ax-7 2031, axc11 2464 } . Almost identical to axc11nfromc11 39562. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	axc11n11r 37170	Proof of axc11n 2460 from { ax-1 6-- ax-7 2031, axc9 2416, axc11r 2402 } (note that axc16 2299 is provable from { ax-1 6-- ax-7 2031, axc11r 2402 }). Note that axc11n 2460 proves (over minimal calculus) that axc11 2464 and axc11r 2402 are equivalent. Therefore, axc11n11 37169 and axc11n11r 37170 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2464 appears slightly stronger since axc11n11r 37170 requires axc9 2416 while axc11n11 37169 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	bj-axc16g16 37171*	Proof of axc16g 2298 from { ax-1 6-- ax-7 2031, axc16 2299 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	bj-ax12v3 37172*	A weak version of ax-12 2215 which is stronger than ax12v 2216. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2035), then bj-ax12v3 37172 implies ax-5 1933 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 37173. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-ax12v3ALT 37173*	Alternate proof of bj-ax12v3 37172. Uses axc11r 2402 and axc15 2456 instead of ax-12 2215. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-sb 37174*	A weak variant of sbid2 2542 not requiring ax-13 2406 nor ax-10 2178. On top of Tarski's FOL, one implication requires only ax12v 2216, and the other requires only sp 2221. (Contributed by BJ, 25-May-2021.)
⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-modalbe 37175	The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2354. (Contributed by BJ, 20-Oct-2019.)
⊢ (𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	bj-spst 37176	Closed form of sps 2223. Once in main part, prove sps 2223 and spsd 2225 from it. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-19.21bit 37177	Closed form of 19.21bi 2227. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
Theorem	bj-19.23bit 37178	Closed form of 19.23bi 2229. (Contributed by BJ, 20-Oct-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓))
Theorem	bj-nexrt 37179	Closed form of nexr 2230. Contrapositive of 19.8a 2219. (Contributed by BJ, 20-Oct-2019.)
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
Theorem	bj-alrim 37180	Closed form of alrimi 2251. (Contributed by BJ, 2-May-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-alrim2 37181	Uncurried (imported) form of bj-alrim 37180. (Contributed by BJ, 2-May-2019.)
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓))
Theorem	bj-nfdt0 37182	A theorem close to a closed form of nf5d 2321 and nf5dh 2184. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))
Theorem	bj-nfdt 37183	Closed form of nf5d 2321 and nf5dh 2184. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
Theorem	bj-nexdt 37184	Closed form of nexd 2259. (Contributed by BJ, 20-Oct-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdvt 37185*	Closed form of nexdv 1959. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
Theorem	bj-alexbiex 37186	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 37187	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 37188	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 37189	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 37190	Strengthening 19.9ht 2355 by replacing its consequent with a biconditional (19.9t 2242 does have a biconditional consequent). This propagates. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))
Theorem	bj-hbntbi 37191	Strengthening hbnt 2331 by replacing its consequent with a biconditional. See also hbntg 36166 and hbntal 45127. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 37190. (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
Theorem	bj-biexal1 37192	A general FOL biconditional that generalizes 19.9ht 2355 among others. For this and the following theorems, see also 19.35 1900, 19.21 2245, 19.23 2249. 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 37193	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 37194	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 37195	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 37196	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 37197	A more general instance of the deduction form of hbex 2360. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑦𝜓)    &   ⊢ (𝜓 → (𝜒 → ∀𝑥𝜃))    ⇒   ⊢ (𝜑 → (∃𝑦𝜒 → ∀𝑥∃𝑦𝜃))
Theorem	bj-hbext 37198	Closed form of bj-hbex 37199 and hbex 2360. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓))
Theorem	bj-hbex 37199	A more general instance of hbex 2360. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓)
Theorem	bj-nfalt 37200	Closed form of nfal 2358. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)