P. 369 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36801-36900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prvlem1 36801	An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (Prv 𝜑 → Prv 𝜓)
Theorem	prvlem2 36802	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 36803	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 36804	See the section header comments for the context, as well as the comments for bj-babygodel 36803. 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/ 36803). (Contributed by BJ, 20-Apr-2019.)
⊢ (𝜓 ↔ (Prv 𝜓 → 𝜑))    &   ⊢ (Prv 𝜑 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-godellob 36805	Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 36803 and bj-babylob 36804 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  Adding ax-gen
Theorem	bj-genr 36806	Generalization rule on the right conjunct. See 19.28 2235. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (𝜑 ∧ ∀𝑥𝜓)
Theorem	bj-genl 36807	Generalization rule on the left conjunct. See 19.27 2234. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ∧ 𝜓)
Theorem	bj-genan 36808	Generalization rule on a conjunction. Forward inference associated with 19.26 1871. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ∧ ∀𝑥𝜓)
Theorem	bj-mpgs 36809	From a closed form theorem (the major premise) with an antecedent in the "strong necessity" modality (in the language of modal logic), deduce the inference ⊢ 𝜑 ⇒ ⊢ 𝜓. Strong necessity is stronger than necessity, and equivalent to it when sp 2190 (modal T) is available. Therefore, this theorem is stronger than mpg 1798 when sp 2190 is not available. (Contributed by BJ, 1-Nov-2023.)
⊢ 𝜑    &   ⊢ ((𝜑 ∧ ∀𝑥𝜑) → 𝜓)    ⇒   ⊢ 𝜓
21.19.4.2  Adding ax-4
Theorem	bj-2alim 36810	Closed form of 2alimi 1813. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-2exim 36811	Closed form of 2eximi 1837. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓))
Theorem	bj-alanim 36812	Closed form of alanimi 1817. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥((𝜑 ∧ 𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))
Theorem	bj-2albi 36813	Closed form of 2albii 1821. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))
Theorem	bj-notalbii 36814	Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4332 (103>94), ballotlem2 34646 (2655>2648), bnj1143 34946 (522>519), hausdiag 23589 (2119>2104). (Contributed by BJ, 17-Jul-2021.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)
Theorem	bj-2exbi 36815	Closed form of 2exbii 1850. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓))
Theorem	bj-3exbi 36816	Closed form of 3exbii 1851. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓))
Theorem	bj-sylggt 36817	Stronger form of sylgt 1823, closer to ax-2 7. (Contributed by BJ, 30-Jul-2025.)
⊢ ((𝜑 → ∀𝑥(𝜓 → 𝜒)) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))
Theorem	bj-sylgt2 36818	Uncurried (imported) form of sylgt 1823. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜓 → 𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))
Theorem	bj-alrimg 36819	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 36823. (Contributed by BJ, 9-Dec-2023.)
⊢ ((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓 → 𝜒) → (𝜑 → ∀𝑥𝜒)))
Theorem	bj-alrimd 36820	A slightly more general alrimd 2222. A common usage will have 𝜑 substituted for 𝜓 and 𝜒 substituted for 𝜃, giving a form closer to alrimd 2222. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜃))    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜏))
Theorem	bj-sylget 36821	Dual statement of sylgt 1823. Closed form of bj-sylge 36824. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜒 → 𝜑) → ((∃𝑥𝜑 → 𝜓) → (∃𝑥𝜒 → 𝜓)))
Theorem	bj-sylget2 36822	Uncurried (imported) form of bj-sylget 36821. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜓 → 𝜒)) → (∃𝑥𝜑 → 𝜒))
Theorem	bj-exlimg 36823	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 36819. (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → (∃𝑥𝜒 → 𝜓)))
Theorem	bj-sylge 36824	Dual statement of sylg 1824 (the final "e" in the label stands for "existential (version of sylg 1824)". Variant of exlimih 2295. (Contributed by BJ, 25-Dec-2023.)
⊢ (∃𝑥𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimd 36825	A slightly more general exlimd 2225. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2225. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (∃𝑥𝜃 → 𝜏))    &   ⊢ (𝜓 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (∃𝑥𝜒 → 𝜏))
Theorem	bj-nfimexal 36826	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 1840) and the converse implication is the join of instances of bj-alrimg 36819 and bj-exlimg 36823 (see 19.38a 1841 and 19.38b 1842). 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 36827	Closed form of aleximi 1833. Note: this proof is shorter, so aleximi 1833 could be deduced from it (exim 1835 would have to be proved first, see bj-eximALT 36841 but its proof is shorter (currently almost a subproof of aleximi 1833)). (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))
Theorem	bj-nexdh 36828	Closed form of nexdh 1866 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdh2 36829	Uncurried (imported) form of bj-nexdh 36828. (Contributed by BJ, 6-May-2019.)
⊢ ((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))
Theorem	bj-hbxfrbi 36830	Closed form of hbxfrbi 1826. Note: it is less important than nfbiit 1852. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 36942) in order not to require sp 2190 (modal T). See bj-hbyfrbi 36831 for its version with existential quantifiers. (Contributed by BJ, 6-May-2019.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
Theorem	bj-hbyfrbi 36831	Version of bj-hbxfrbi 36830 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓)))
Theorem	bj-exalim 36832	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 1911. I propose to move to the main part: bj-exalim 36832, bj-exalimi 36833, bj-exalims 36834, bj-exalimsi 36835, bj-ax12i 36837, bj-ax12wlem 36844, bj-ax12w 36878. A new label is needed for bj-ax12i 36837 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to ∃𝑥 in speimfw 1964 and spimfw 1966 (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 36833	An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 36832 (using mpg 1798) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1964 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))
Theorem	bj-exalims 36834	Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1966 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)))
Theorem	bj-exalimsi 36835	An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1966 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-ax12ig 36836	A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 36837. (Contributed by BJ, 19-Dec-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-ax12i 36837	A weakening of bj-ax12ig 36836 that is sufficient to prove a weak form of the axiom of substitution ax-12 2184. The general statement of which ax12i 1967 is an instance. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-nfimt 36838	Closed form of nfim 1897 and curried (exported) form of nfimt 1896. (Contributed by BJ, 20-Oct-2021.)
⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓)))
Theorem	bj-cbvalimt 36839	A lemma in closed form used to prove bj-cbval 36849 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1878 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 36840	A lemma in closed form used to prove bj-cbvex 36850 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1878 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 36841	Alternate proof of exim 1835 directly from alim 1811 by using df-ex 1781 (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 36842	Alternate proof of aleximi 1833 from exim 1835, 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 36843	A commuted form of exim 1835 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 36844*	A lemma used to prove a weak version of the axiom of substitution ax-12 2184. (Temporary comment: The general statement that ax12wlem 2137 proves.) (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-cbvalim 36845*	A lemma used to prove bj-cbval 36849 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 36846*	A lemma used to prove bj-cbvex 36850 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 36847*	An equality-free general instance of one half of a precise form of bj-cbval 36849. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ ∀𝑦∃𝑥𝜒    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	bj-cbveximi 36848*	An equality-free general instance of one half of a precise form of bj-cbvex 36850. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ ∀𝑥∃𝑦𝜒    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbval 36849*	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 36850*	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 36851	Syntax for BJ's version of the uniqueness quantifier.
wff ∃**𝑥𝜑
Definition	df-bj-mo 36852*	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-df-sb 36853*	Proposed definition to replace df-sb 2068 and df-sbc 3741. Proof is therefore unimportant. Contrary to df-sb 2068, 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. The equality 𝑦 = 𝑥 may seem "reversed", but it is written this way so that "substitution for oneself" does not require symmetry of equality to be seen to be the identity on propositions. (Contributed by BJ, 19-Feb-2026.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑦 = 𝑥 → 𝜑)))
Theorem	bj-ssbeq 36854*	Substitution in an equality, disjoint variables case. Uses only ax-1 6 through ax-6 1968. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 36854 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 36855*	A lemma for the definiens of df-sb 2068. An instance of sp 2190 proved without it. Note: it has a common subproof with sbjust 2066. (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 36856*	An instance of ax-11 2162 proved without it. The converse may not be provable without ax-11 2162 (since using alcomimw 2044 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 36857*	A weaker form of ax-12 2184 and ax12v 2185, 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 36858*	Remove a DV condition from bj-ax12v 36857 (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 36859*	Axiom bj-ax12 36858 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑)
Theorem	bj-19.41al 36860	Special case of 19.41 2242 proved from core axioms, ax-10 2146 (modal5), and hba1 2299 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	bj-equsexval 36861*	Special case of equsexv 2275 proved from core axioms, ax-10 2146 (modal5), and hba1 2299 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓)
Theorem	bj-subst 36862*	Proof of sbalex 2249 from core axioms, ax-10 2146 (modal5), and bj-ax12 36858. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	bj-ssbid2 36863	A special case of sbequ2 2256. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid2ALT 36864	Alternate proof of bj-ssbid2 36863, not using sbequ2 2256. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
Theorem	bj-ssbid1 36865	A special case of sbequ1 2255. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ssbid1ALT 36866	Alternate proof of bj-ssbid1 36865, not using sbequ1 2255. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ax6elem1 36867*	Lemma for bj-ax6e 36869. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	bj-ax6elem2 36868*	Lemma for bj-ax6e 36869. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Theorem	bj-ax6e 36869	Proof of ax6e 2387 (hence ax6 2388) from Tarski's system, ax-c9 39146, ax-c16 39148. Remark: ax-6 1968 is used only via its principal (unbundled) instance ax6v 1969. (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 36870*	Closed form of spimvw 1987. See also spimt 2390. (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-spnfw 36871	Theorem close to a closed form of spnfw 1980. (Contributed by BJ, 12-May-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbvexiw 36872*	Change bound variable. This is to cbvexvw 2038 what cbvaliw 2007 is to cbvalvw 2037. TODO: move after cbvalivw 2008. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbvexivw 36873*	Change bound variable. This is to cbvexvw 2038 what cbvalivw 2008 is to cbvalvw 2037. TODO: move after cbvalivw 2008. (Contributed by BJ, 17-Mar-2020.)
⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-modald 36874	A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	bj-denot 36875*	A weakening of ax-6 1968 and ax6v 1969. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
⊢ (𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)
Theorem	bj-eqs 36876*	A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2376. 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 36877*	Change bound variable. This is to cbvexvw 2038 what cbvalw 2036 is to cbvalvw 2037. (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 36878*	The general statement that ax12w 2138 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 36879	A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2115 and ax-9 2123. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2115 and ax-9 2123, as proved here. In the other direction, one can prove ax-8 2115 (respectively ax-9 2123) from bj-ax89 36879 by using mpan2 691 (respectively mpan 690) and equid 2013. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))
Theorem	bj-cleljusti 36880*	One direction of cleljust 2122, requiring only ax-1 6-- ax-5 1911 and ax8v1 2117. (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 36881	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 1810 section, soon after 2nexaln 1831, and used to prove excom 2167. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))
Theorem	bj-hbalt 36882	Closed form of hbal 2172. When in main part, prove hbal 2172 and hbald 2173 from it. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
21.19.4.9  Adding ax-12
Theorem	axc11n11 36883	Proof of axc11n 2430 from { ax-1 6-- ax-7 2009, axc11 2434 } . Almost identical to axc11nfromc11 39182. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	axc11n11r 36884	Proof of axc11n 2430 from { ax-1 6-- ax-7 2009, axc9 2386, axc11r 2372 } (note that axc16 2268 is provable from { ax-1 6-- ax-7 2009, axc11r 2372 }). Note that axc11n 2430 proves (over minimal calculus) that axc11 2434 and axc11r 2372 are equivalent. Therefore, axc11n11 36883 and axc11n11r 36884 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2434 appears slightly stronger since axc11n11r 36884 requires axc9 2386 while axc11n11 36883 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	bj-axc16g16 36885*	Proof of axc16g 2267 from { ax-1 6-- ax-7 2009, axc16 2268 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	bj-ax12v3 36886*	A weak version of ax-12 2184 which is stronger than ax12v 2185. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2013), then bj-ax12v3 36886 implies ax-5 1911 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 36887. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-ax12v3ALT 36887*	Alternate proof of bj-ax12v3 36886. Uses axc11r 2372 and axc15 2426 instead of ax-12 2184. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-sb 36888*	A weak variant of sbid2 2512 not requiring ax-13 2376 nor ax-10 2146. On top of Tarski's FOL, one implication requires only ax12v 2185, and the other requires only sp 2190. (Contributed by BJ, 25-May-2021.)
⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-modalbe 36889	The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2324. (Contributed by BJ, 20-Oct-2019.)
⊢ (𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	bj-spst 36890	Closed form of sps 2192. Once in main part, prove sps 2192 and spsd 2194 from it. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-19.21bit 36891	Closed form of 19.21bi 2196. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
Theorem	bj-19.23bit 36892	Closed form of 19.23bi 2198. (Contributed by BJ, 20-Oct-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓))
Theorem	bj-nexrt 36893	Closed form of nexr 2199. Contrapositive of 19.8a 2188. (Contributed by BJ, 20-Oct-2019.)
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
Theorem	bj-alrim 36894	Closed form of alrimi 2220. (Contributed by BJ, 2-May-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-alrim2 36895	Uncurried (imported) form of bj-alrim 36894. (Contributed by BJ, 2-May-2019.)
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓))
Theorem	bj-nfdt0 36896	A theorem close to a closed form of nf5d 2290 and nf5dh 2152. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))
Theorem	bj-nfdt 36897	Closed form of nf5d 2290 and nf5dh 2152. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
Theorem	bj-nexdt 36898	Closed form of nexd 2228. (Contributed by BJ, 20-Oct-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdvt 36899*	Closed form of nexdv 1937. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
Theorem	bj-alexbiex 36900	Adding a second quantifier over the same variable is a transparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)