P. 371 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37001-37100   *Has distinct variable group(s)

Type	Label	Description
Statement
Axiom	ax-prv1 37001	First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ 𝜑    ⇒   ⊢ Prv 𝜑
Axiom	ax-prv2 37002	Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
Axiom	ax-prv3 37003	Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (Prv 𝜑 → Prv Prv 𝜑)
Theorem	prvlem1 37004	An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (Prv 𝜑 → Prv 𝜓)
Theorem	prvlem2 37005	An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))
Theorem	bj-babygodel 37006	See the section header comments for the context. The first hypothesis reads "𝜑 is true if and only if it is not provable in T" (and having this first hypothesis means that we can prove this fact in T). The wff 𝜑 is a formal version of the sentence "This sentence is not provable". The hard part of the proof of Gödel's theorem is to construct such a 𝜑, called a "Gödel–Rosser sentence", for a first-order theory T which is effectively axiomatizable and contains Robinson arithmetic, through Gödel diagonalization (this can be done in primitive recursive arithmetic). The second hypothesis means that ⊥ is not provable in T, that is, that the theory T is consistent (and having this second hypothesis means that we can prove in T that the theory T is consistent). The conclusion is the falsity, so having the conclusion means that T can prove the falsity, that is, T is inconsistent. Therefore, taking the contrapositive, this theorem expresses that if a first-order theory is consistent (and one can prove in it that some formula is true if and only if it is not provable in it), then this theory does not prove its own consistency. This proof is due to George Boolos, Gödel's Second Incompleteness Theorem Explained in Words of One Syllable, Mind, New Series, Vol. 103, No. 409 (January 1994), pp. 1--3. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝜑 ↔ ¬ Prv 𝜑)    &   ⊢ ¬ Prv ⊥    ⇒   ⊢ ⊥
Theorem	bj-babylob 37007	See the section header comments for the context, as well as the comments for bj-babygodel 37006. Löb's theorem when the Löb sentence is given as a hypothesis (the hard part of the proof of Löb's theorem is to construct this Löb sentence; this can be done, using Gödel diagonalization, for any first-order effectively axiomatizable theory containing Robinson arithmetic). More precisely, the present theorem states that if a first-order theory proves that the provability of a given sentence entails its truth (and if one can construct in this theory a provability predicate and a Löb sentence, given here as the first hypothesis), then the theory actually proves that sentence. See for instance, Eliezer Yudkowsky, The Cartoon Guide to Löb's Theorem (available at http://yudkowsky.net/rational/lobs-theorem/ 37006). (Contributed by BJ, 20-Apr-2019.)
⊢ (𝜓 ↔ (Prv 𝜓 → 𝜑))    &   ⊢ (Prv 𝜑 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-godellob 37008	Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 37006 and bj-babylob 37007 for details). (Contributed by BJ, 20-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ↔ ¬ Prv 𝜑)    &   ⊢ ¬ Prv ⊥    ⇒   ⊢ ⊥
21.19.4  First-order logic Utility lemmas or strengthenings of theorems in the main part (biconditional or closed forms, or fewer disjoint variable conditions, or disjoint variable conditions replaced with nonfreeness hypotheses...). Sorted in the same order as in the main part.
21.19.4.1  Universal and existential quantifiers, nonfreeness predicate
Theorem	bj-exexalal 37009	A lemma for changing bound variables. Only the forward implication is intuitionistic. (Contributed by BJ, 14-Mar-2026.)
⊢ ((∃𝑥𝜑 → ∃𝑦𝜓) ↔ (∀𝑦 ¬ 𝜓 → ∀𝑥 ¬ 𝜑))
21.19.4.2  Adding ax-gen
Theorem	bj-genr 37010	Generalization rule on the right conjunct. See 19.28 2262. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (𝜑 ∧ ∀𝑥𝜓)
Theorem	bj-genl 37011	Generalization rule on the left conjunct. See 19.27 2261. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ∧ 𝜓)
Theorem	bj-genan 37012	Generalization rule on a conjunction. Forward inference associated with 19.26 1889. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ∧ ∀𝑥𝜓)
Theorem	bj-mpgs 37013	From a closed form theorem (the major premise) with an antecedent in the "strong necessity" modality (in the language of modal logic), deduce the associated inference. Strong necessity is stronger than necessity, and equivalent to it when sp 2217 (modal T) is available. Therefore, this theorem is stronger than mpg 1816, and strictly stronger when sp 2217 is not available. (Contributed by BJ, 1-Nov-2023.)
⊢ ((𝜑 ∧ ∀𝑥𝜑) → 𝜓)    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
21.19.4.3  Adding ax-4
Theorem	bj-almp 37014	A quantified form of ax-mp 5. See also barbara 2688, bj-ala1i 37021, bj-almpi 37022. (Contributed by BJ, 19-Mar-2026.)
⊢ ∀𝑥(𝜓 → 𝜑)    &   ⊢ ∀𝑥𝜓    ⇒   ⊢ ∀𝑥𝜑
Theorem	bj-sylggt 37015	Stronger form of sylgt 1841, closer to ax-2 7. (Contributed by BJ, 30-Jul-2025.)
⊢ ((𝜑 → ∀𝑥(𝜓 → 𝜒)) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))
Theorem	bj-alrimg 37016	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 37038. (Contributed by BJ, 9-Dec-2023.)
⊢ ((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓 → 𝜒) → (𝜑 → ∀𝑥𝜒)))
Theorem	bj-sylgt2 37017	Uncurried (imported) form of sylgt 1841. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜓 → 𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))
Theorem	bj-nexdh 37018	Closed form of nexdh 1884 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdh2 37019	Uncurried (imported) form of bj-nexdh 37018. (Contributed by BJ, 6-May-2019.)
⊢ ((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))
Theorem	bj-alimii 37020	Inference associated with alimi 1830. Double inference associated with alim 1829. The usual proof of an associated inference (here from alimi 1830 and ax-mp 5) has the same size and same number of steps. (Contributed by BJ, 19-Mar-2026.)
⊢ (𝜓 → 𝜑)    &   ⊢ ∀𝑥𝜓    ⇒   ⊢ ∀𝑥𝜑
Theorem	bj-ala1i 37021	Add an antecedent in a universally quantified formula. Inference associated with ala1 1832. (Contributed by BJ, 6-Oct-2018.)
⊢ ∀𝑥𝜑    ⇒   ⊢ ∀𝑥(𝜓 → 𝜑)
Theorem	bj-almpi 37022	A quantified form of mpi 20. See also barbara 2688, bj-ala1i 37021, bj-almp 37014. (Contributed by BJ, 19-Mar-2026.)
⊢ ∀𝑥(𝜑 → (𝜒 → 𝜓))    &   ⊢ ∀𝑥𝜒    ⇒   ⊢ ∀𝑥(𝜑 → 𝜓)
Theorem	bj-almpig 37023	A partially quantified form of mpi 20 similar to bj-almpi 37022. (Contributed by BJ, 19-Mar-2026.)
⊢ (𝜑 → (𝜒 → 𝜓))    &   ⊢ ∀𝑥𝜒    ⇒   ⊢ ∀𝑥(𝜑 → 𝜓)
Theorem	bj-alsyl 37024	Syllogism under the universal quantifier, in the curried form appearing as Theorem *10.3 of [WhiteheadRussell] p. 145. See alsyl 1912 for the uncurried form. (Contributed by BJ, 28-Mar-2026.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜒)))
Theorem	bj-2alim 37025	Closed form of 2alimi 1831. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-alimdh 37026	General instance of alimdh 1836. (Contributed by NM, 4-Jan-2002.) State the most general derivable instance. (Revised by BJ, 5-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜓 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑥𝜃))
Theorem	bj-alrimdh 37027	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2241 and 19.21h 2320. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) State the most general derivable instance. (Revised by BJ, 5-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜒 → ∀𝑥𝜃)    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜏))
Theorem	bj-alrimd 37028	A slightly more general alrimd 2249. A common usage will have 𝜑 substituted for 𝜓 and 𝜒 substituted for 𝜃, giving a form closer to alrimd 2249. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜃))    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜏))
Theorem	bj-exa1i 37029	Add an antecedent in an existentially quantified formula. Inference associated with exa1 1857. (Contributed by BJ, 6-Oct-2018.)
⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜓 → 𝜑)
Theorem	bj-alanim 37030	Closed form of alanimi 1835. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥((𝜑 ∧ 𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))
Theorem	bj-2albi 37031	Closed form of 2albii 1839. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))
Theorem	bj-notalbii 37032	Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4330 (103>94), ballotlem2 34746 (2655>2648), bnj1143 35045 (522>519), hausdiag 23692 (2119>2104). (Contributed by BJ, 17-Jul-2021.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)
Theorem	bj-2exim 37033	Closed form of 2eximi 1855. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓))
Theorem	bj-2exbi 37034	Closed form of 2exbii 1868. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓))
Theorem	bj-3exbi 37035	Closed form of 3exbii 1869. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓))
Theorem	bj-sylget 37036	Dual statement of sylgt 1841. Closed form of bj-sylge 37039. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜒 → 𝜑) → ((∃𝑥𝜑 → 𝜓) → (∃𝑥𝜒 → 𝜓)))
Theorem	bj-sylget2 37037	Uncurried (imported) form of bj-sylget 37036. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜓 → 𝜒)) → (∃𝑥𝜑 → 𝜒))
Theorem	bj-exlimg 37038	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 37016. (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → (∃𝑥𝜒 → 𝜓)))
Theorem	bj-sylge 37039	Dual statement of sylg 1842 (the final "e" in the label stands for "existential (version of sylg 1842)". Variant of exlimih 2322. (Contributed by BJ, 25-Dec-2023.)
⊢ (∃𝑥𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimd 37040	A slightly more general exlimd 2252. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2252. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (∃𝑥𝜃 → 𝜏))    &   ⊢ (𝜓 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∃𝑥𝜒 → 𝜏))
Theorem	bj-nfimexal 37041	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 1858) and the converse implication is the join of instances of bj-alrimg 37016 and bj-exlimg 37038 (see 19.38a 1859 and 19.38b 1860). 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-exim 37042	Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) Prove it directly from alim 1829 to allow use in bj-alexim 37043. (Revised by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-alexim 37043	Closed form of aleximi 1851. Note: this proof is shorter, so aleximi 1851 could be deduced from it (exim 1853 would have to be proved first, see bj-exim 37042). (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))
Theorem	bj-aleximiALT 37044	Alternate proof of aleximi 1851 from exim 1853, 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-hbxfrbi 37045	Closed form of hbxfrbi 1844. Note: it is less important than nfbiit 1870. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 37178) in order not to require sp 2217 (modal T). See bj-hbyfrbi 37046 for its version with existential quantifiers. (Contributed by BJ, 6-May-2019.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
Theorem	bj-hbyfrbi 37046	Version of bj-hbxfrbi 37045 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓)))
Theorem	bj-exalim 37047	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 1929. I propose to move to the main part: bj-exalim 37047, bj-exalimi 37048, bj-eximcom 37049 bj-exalims 37050, bj-exalimsi 37051, bj-ax12i 37054, bj-ax12wlem 37077, bj-ax12w 37110. A new label is needed for bj-ax12i 37054 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to ∃𝑥 in speimfw 1982 and spimfw 1984 (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 37048	An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 37047 (using mpg 1816) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1982 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))
Theorem	bj-eximcom 37049	A commuted form of exim 1853 which is sometimes posited as an axiom in instuitionistic modal logic. Forward implication of 19.35 1896. Its converse is not intuitionistic. (Contributed by BJ, 9-Dec-2023.)
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-exalims 37050	Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1984 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)))
Theorem	bj-exalimsi 37051	An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1984 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-axdd2ALT 37052	Alternate proof of bj-axdd2 36995 (this should replace bj-axdd2 36995 when bj-exalimi 37048 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 37053	A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 37054. (Contributed by BJ, 19-Dec-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-ax12i 37054	A weakening of bj-ax12ig 37053 that is sufficient to prove a weak form of the axiom of substitution ax-12 2211. The general statement of which ax12i 1985 is an instance. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-nfimt 37055	Closed form of nfim 1915 and curried (exported) form of nfimt 1914. (Contributed by BJ, 20-Oct-2021.) Proof should not use 19.35 1896. (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓)))
Theorem	bj-spimnfe 37056	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 2265, bj-19.36im 37198, 19.36imv 1964, spimfw 1984... (Contributed by BJ, 3-Apr-2026.) Proof should not use 19.35 1896. (Proof modification is discouraged.)
⊢ ((∃𝑥𝜓 → 𝜓) → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-spimenfa 37057	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 1896. (Proof modification is discouraged.)
⊢ ((𝜑 → ∀𝑥𝜑) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))
Theorem	bj-spim 37058	A lemma for universal specification. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1988 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 37059	A lemma for existential generalization. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1988 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 37060	A lemma for alpha-renaming of variables bound by a universal quantifier. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1988 will prove Hypothesis bj-cbvalimd0.denote. When ax6ev 1988 is not available but only its universal closure is, then bj-cbvalimd 37063 or bj-cbvalimdv 37065 should be used (see bj-cbvalimdlem 37061, bj-cbval 37078). (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 37061	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 37065 or from a nonfreeness condition and alcom 2192 as in bj-cbvalimd 37063. Hypothesis bj-cbvalimdlem.denote is weaker than the corresponding hypothesis of bj-cbvalimd0 37060, and this proof is therefore a bit longer, not using bj-spim 37058 but bj-eximcom 37049. (Contributed by BJ, 12-Mar-2023.) Proof should not use 19.35 1896. (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 37062	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 37066 or from a nonfreeness condition and excom 2195 as in bj-cbveximd 37064. 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 37059 but bj-eximcom 37049. (Contributed by BJ, 12-Mar-2023.) Proof should not use 19.35 1896. (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 37063	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 37064	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 37065*	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 37066*	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 37067*	Version of spvw 2000 and 19.3v 2001 proved from ax-1 6-- ax-5 1929. The antecedent can for instance be proved with the existence axiom extru 1994. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (𝜓 ↔ ∀𝑥𝜓))
Theorem	bj-spvew 37068*	Version of 19.8v 2002 and 19.9v 2003 proved from ax-1 6-- ax-5 1929. The antecedent can for instance be proved with the existence axiom extru 1994. (Contributed by BJ, 8-Mar-2026.) This could also be proved from bj-spvw 37067 using duality, but that proof would not be intuitionistic, contrary to the present one. (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (𝜓 ↔ ∃𝑥𝜓))
Theorem	bj-alextruim 37069*	An equivalent expression for universal quantification over a non-occurring variable proved over ax-1 6-- ax-5 1929. The forward implication can be strengthened when ax-6 1986 is posited (which implies that models are non-empty), see spvw 2000. The reverse implication can be seen as a strengthening of ax-5 1929 (since the antecedent of the implication is weakened). See bj-exextruan 37070 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 37070*	An equivalent expression for existential quantification over a non-occurring variable proved over ax-1 6-- ax-5 1929. The forward implication can be seen as a strengthening of ax-5 1929 (a conjunct is added to the consequent of the implication). The reverse implication can be strengthened when ax-6 1986 is posited (which implies that models are non-empty), see 19.8v 2002. See bj-alextruim 37069 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 37071*	Universally quantifying over a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1929 and the existence axiom extru 1994. See bj-cbvaw 37073 for a strengthening. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∀𝑦𝜓))
Theorem	bj-cbvexvv 37072*	Existentially quantifying over a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1929 and the existence axiom extru 1994. See bj-cbvew 37074 for a strengthening. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (∃𝑦𝜓 → ∃𝑥𝜓))
Theorem	bj-cbvaw 37073*	Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 37071. 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 187); an intuitionistically valid statement is obtained by expressing the antecedent as a disjunction (classically equivalent through imor 864). (Proof modification is discouraged.)
⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∀𝑥𝜓 → ∀𝑦𝜓))
Theorem	bj-cbvew 37074*	Existentially quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvexvv 37072. 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 37075*	Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 37071. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ((∃𝑥⊤ → ∃𝑦𝜑) → (∀𝑦𝜓 → ∀𝑥𝜓))
Theorem	bj-cbvaew 37076*	Exixtentially quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvexvv 37072. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∃𝑦𝜓 → ∃𝑥𝜓))
Theorem	bj-ax12wlem 37077*	A lemma used to prove a weak version of the axiom of substitution ax-12 2211. (Temporary comment: The general statement that ax12wlem 2165 proves.) (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-cbval 37078*	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 1929. (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 37079*	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 1929. (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 37080	Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑
Definition	df-bj-mo 37081*	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 37082*	Proposed definition to replace df-sb 2090 and df-sbc 3743. Proof is therefore unimportant. Contrary to df-sb 2090, 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 1988 is posited, all variables "exist". (Contributed by BJ, 19-Feb-2026.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-sbcex 37083	Proof of sbcex 3752 when taking bj-df-sb 37082 as definition. (Contributed by BJ, 19-Feb-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
Theorem	bj-dfsbc 37084	Proof of df-sbc 3743 when taking bj-df-sb 37082 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 37085*	Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 1986. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 37085 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 37086*	A lemma for the definiens of df-sb 2090. An instance of sp 2217 proved without it. Note: it has a common subproof with rename-sb 2088. (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 37087*	An instance of ax-11 2190 proved without it. The converse may not be provable without ax-11 2190 (since using alcomimw 2062 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 37088*	A weaker form of ax-12 2211 and ax12v 2212, 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 37089*	Remove a DV condition from bj-ax12v 37088 (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 37090*	Axiom bj-ax12 37089 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑)
Theorem	bj-19.41al 37091	Special case of 19.41 2269 proved from core axioms, ax-10 2174 (modal5), and hba1 2326 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	bj-equsexval 37092*	Special case of equsexv 2302 proved from core axioms, ax-10 2174 (modal5), and hba1 2326 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓)
Theorem	bj-subst 37093*	Proof of sbalex 2276 from core axioms, ax-10 2174 (modal5), and bj-ax12 37089. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	bj-ssbid2 37094	A special case of sbequ2 2283. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid2ALT 37095	Alternate proof of bj-ssbid2 37094, not using sbequ2 2283. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid1 37096	A special case of sbequ1 2282. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ssbid1ALT 37097	Alternate proof of bj-ssbid1 37096, not using sbequ1 2282. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ax6elem1 37098*	Lemma for bj-ax6e 37100. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	bj-ax6elem2 37099*	Lemma for bj-ax6e 37100. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Theorem	bj-ax6e 37100	Proof of ax6e 2413 (hence ax6 2414) from Tarski's system, ax-c9 39474, ax-c16 39476. Remark: ax-6 1986 is used only via its principal (unbundled) instance ax6v 1987. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦