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Theorem List for Metamath Proof Explorer - 34101-34200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-nnfnth 34101 A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.)
¬ 𝜑       Ⅎ'𝑥𝜑

Theorembj-nnfim1 34102 A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))

Theorembj-nnfim2 34103 A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑𝜓)))

Theorembj-nnfim 34104 Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication. (Contributed by BJ, 27-Aug-2023.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))

Theorembj-nnfimd 34105 Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication, deduction form. (Contributed by BJ, 2-Dec-2023.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))

Theorembj-nnfan 34106 Nonfreeness in both conjuncts implies nonfreeness in the conjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of conjunction in terms of implication and negation, so using bj-nnfim 34104, bj-nnfnt 34098 and bj-nnfbi 34086, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))

Theorembj-nnfand 34107 Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 34106, it has two more essential steps but fewer total steps (since there are fewer intermediate formulas to build) and is easier to follow and understand. This statement is of intermediate complexity: for simpler statements, closed-style proofs like that of bj-nnfan 34106 will generally be shorter than deduction-style proofs while still easy to follow, while for more complex statements, the opposite will be true (and deduction-style proofs like that of bj-nnfand 34107 will generally be easier to understand). (Contributed by BJ, 19-Nov-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))

Theorembj-nnfor 34108 Nonfreeness in both disjuncts implies nonfreeness in the disjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of disjunction in terms of implication and negation, so using bj-nnfim 34104, bj-nnfnt 34098 and bj-nnfbi 34086, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))

Theorembj-nnford 34109 Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 34108 and bj-nnfand 34107. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))

Theorembj-nnfbit 34110 Nonfreeness in both sides implies nonfreeness in the biconditional. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))

Theorembj-nnfbid 34111 Nonfreeness in both sides implies nonfreeness in the biconditional, deduction form. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))

Theorembj-nnfv 34112* A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.)
Ⅎ'𝑥𝜑

Theorembj-nnf-alrim 34113 Proof of the closed form of alrimi 2215 from modalK (compare alrimiv 1929). See also bj-alrim 34054. Actually, most proofs between 19.3t 2203 and 2sbbid 2249 could be proved without ax-12 2179. (Contributed by BJ, 20-Aug-2023.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Theorembj-nnf-exlim 34114 Proof of the closed form of exlimi 2219 from modalK (compare exlimiv 1932). See also bj-sylget2 33982. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓)))

Theorembj-dfnnf3 34115 Alternate definition of nonfreeness when sp 2184 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1786. (Proof modification is discouraged.)
(Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Theorembj-nfnnfTEMP 34116 New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2184. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1786 except via df-nf 1786 directly. (Proof modification is discouraged.)
(Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)

Theorembj-nnfa1 34117 See nfa1 2156. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
Ⅎ'𝑥𝑥𝜑

Theorembj-nnfe1 34118 See nfe1 2155. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
Ⅎ'𝑥𝑥𝜑

Theorembj-19.12 34119 See 19.12 2348. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2170 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1786 or df-bj-nnf 34085, directly or indirectly. (Proof modification is discouraged.)
(∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theorembj-nnflemaa 34120 One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 34042. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝑥𝜑))

Theorembj-nnflemee 34121 One of four lemmas for nonfreeness: antecedent and consequent both expressed using existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥(∃𝑦𝜑𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝜑))

Theorembj-nnflemae 34122 One of four lemmas for nonfreeness: antecedent expressed with universal quantifier and consequent expressed with existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦𝑥𝜑))

Theorembj-nnflemea 34123 One of four lemmas for nonfreeness: antecedent expressed with existential quantifier and consequent expressed with universal quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥(∃𝑦𝜑𝜑) → (∃𝑦𝑥𝜑 → ∀𝑥𝜑))

Theorembj-nnfalt 34124 See nfal 2344 and bj-nfalt 34072. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦𝑥𝜑)

Theorembj-nnfext 34125 See nfex 2345 and bj-nfext 34073. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
(∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦𝑥𝜑)

Theorembj-stdpc5t 34126 Alias of bj-nnf-alrim 34113 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2210 proved from modalK (obsoleting stdpc5v 1940). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 34113 instead. (New usaged is discouraged.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Theorembj-19.21t 34127 Statement 19.21t 2208 proved from modalK (obsoleting 19.21v 1941). (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theorembj-19.23t 34128 Statement 19.23t 2212 proved from modalK (obsoleting 19.23v 1944). (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Theorembj-19.36im 34129 One direction of 19.36 2234 from the same axioms as 19.36imv 1947. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓)))

Theorembj-19.37im 34130 One direction of 19.37 2236 from the same axioms as 19.37imv 1949. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓)))

Theorembj-19.42t 34131 Closed form of 19.42 2240 from the same axioms as 19.42v 1955. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))

Theorembj-19.41t 34132 Closed form of 19.41 2239 from the same axioms as 19.41v 1951. The same is doable with 19.27 2231, 19.28 2232, 19.31 2238, 19.32 2237, 19.44 2241, 19.45 2242. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Theorembj-sbft 34133 Version of sbft 2272 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.)
(Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑𝜑))

Theorembj-axc10 34134 Alternate (shorter) proof of axc10 2405. One can prove a version with DV (𝑥, 𝑦) without ax-13 2392, by using ax6ev 1973 instead of ax6e 2403. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Theorembj-alequex 34135 A fol lemma. See alequexv 2008 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2406 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)

Theorembj-spimt2 34136 A step in the proof of spimt 2406. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ((∃𝑥𝜓𝜓) → (∀𝑥𝜑𝜓)))

Theorembj-cbv3ta 34137 Closed form of cbv3 2417. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦(∃𝑥𝜓𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Theorembj-cbv3tb 34138 Closed form of cbv3 2417. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦𝑥𝜓 ∧ ∀𝑥𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Theorembj-hbsb3t 34139 A theorem close to a closed form of hbsb3 2528. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Theorembj-hbsb3 34140 Shorter proof of hbsb3 2528. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-nfs1t 34141 A theorem close to a closed form of nfs1 2529. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theorembj-nfs1t2 34142 A theorem close to a closed form of nfs1 2529. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theorembj-nfs1 34143 Shorter proof of nfs1 2529 (three essential steps instead of four). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
𝑦𝜑       𝑥[𝑦 / 𝑥]𝜑

20.15.4.12  Removing dependencies on ax-13 (and ax-11)

It is known that ax-13 2392 is logically redundant (see ax13w 2141 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2392 from every theorem in set.mm which is totally unbundled (i.e., has disjoint variable conditions on all setvar variables). Indeed, start with the existing proof, and replace any occurrence of ax-13 2392 with ax13w 2141.

This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2392 (and using ax6v 1972 / ax6ev 1973 instead of ax-6 1971 / ax6e 2403, as is currently done).

One reason to be optimistic is that the first few utility theorems using ax-13 2392 (roughly 200 of them) are then used mainly with dummy variables, which one can assume distinct from any other, so that the unbundled versions of the utility theorems suffice.

In this section, we prove versions of theorems in the main part with dv conditions and not requiring ax-13 2392, labeled bj-xxxv (we follow the proof of xxx but use ax6v 1972 and ax6ev 1973 instead of ax-6 1971 and ax6e 2403, and ax-5 1912 instead of ax13v 2393; shorter proofs may be possible). When no additional dv condition is required, we label it bj-xxx.

It is important to keep all the bundled theorems already in set.mm, but one may also add the (partially) unbundled versions which dipense with ax-13 2392, so as to remove dependencies on ax-13 2392 from many existing theorems.

UPDATE: it turns out that several theorems of the form bj-xxxv, or minor variations, are already in set.mm with label xxxw.

It is also possible to remove dependencies on ax-11 2162, typically by replacing a nonfree hypothesis with a disjoint variable condition (see cbv3v2 2245 and following theorems).

Theorembj-axc10v 34144* Version of axc10 2405 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Theorembj-spimtv 34145* Version of spimt 2406 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))

Theorembj-cbv3hv2 34146* Version of cbv3h 2426 with two disjoint variable conditions, which does not require ax-11 2162 nor ax-13 2392. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theorembj-cbv1hv 34147* Version of cbv1h 2427 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓 → ∀𝑦𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theorembj-cbv2hv 34148* Version of cbv2h 2428 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓 → ∀𝑦𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theorembj-cbv2v 34149* Version of cbv2 2425 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theorembj-cbvaldv 34150* Version of cbvald 2430 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theorembj-cbvexdv 34151* Version of cbvexd 2431 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theorembj-cbval2vv 34152* Version of cbval2vv 2437 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theorembj-cbvex2vv 34153* Version of cbvex2vv 2438 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theorembj-cbvaldvav 34154* Version of cbvaldva 2432 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theorembj-cbvexdvav 34155* Version of cbvexdva 2433 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theorembj-cbvex4vv 34156* Version of cbvex4v 2439 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))    &   ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))       (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)

Theorembj-equsalhv 34157* Version of equsalh 2444 with a disjoint variable condition, which does not require ax-13 2392. Remark: this is the same as equsalhw 2301. TODO: delete after moving the following paragraph somewhere.

Remarks: equsexvw 2012 has been moved to Main; the theorem ax13lem2 2396 has a dv version which is a simple consequence of ax5e 1914; the theorems nfeqf2 2397, dveeq2 2398, nfeqf1 2399, dveeq1 2400, nfeqf 2401, axc9 2402, ax13 2395, have dv versions which are simple consequences of ax-5 1912. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theorembj-axc11nv 34158* Version of axc11n 2450 with a disjoint variable condition; instance of aevlem 2061. TODO: delete after checking surrounding theorems. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theorembj-aecomsv 34159* Version of aecoms 2452 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2453 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5259). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theorembj-axc11v 34160* Version of axc11 2454 with a disjoint variable condition, which does not require ax-13 2392 nor ax-10 2146. Remark: the following theorems (hbae 2455, nfae 2457, hbnae 2456, nfnae 2458, hbnaes 2459) would need to be totally unbundled to be proved without ax-13 2392, hence would be simple consequences of ax-5 1912 or nfv 1916. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theorembj-drnf2v 34161* Version of drnf2 2468 with a disjoint variable condition, which does not require ax-10 2146, ax-11 2162, ax-12 2179, ax-13 2392. Instance of nfbidv 1924. Note that the version of axc15 2446 with a disjoint variable condition is actually ax12v2 2181 (up to adding a superfluous antecedent). (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Theorembj-equs45fv 34162* Version of equs45f 2484 with a disjoint variable condition, which does not require ax-13 2392. Note that the version of equs5 2485 with a disjoint variable condition is actually sb56 2279 (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
𝑦𝜑       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-hbs1 34163* Version of hbsb2 2523 with a disjoint variable condition, which does not require ax-13 2392, and removal of ax-13 2392 from hbs1 2276. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-nfs1v 34164* Version of nfsb2 2524 with a disjoint variable condition, which does not require ax-13 2392, and removal of ax-13 2392 from nfs1v 2161. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
𝑥[𝑦 / 𝑥]𝜑

Theorembj-hbsb2av 34165* Version of hbsb2a 2525 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-hbsb3v 34166* Version of hbsb3 2528 with a disjoint variable condition, which does not require ax-13 2392. (Remark: the unbundled version of nfs1 2529 is given by bj-nfs1v 34164.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-nfsab1 34167* Remove dependency on ax-13 2392 from nfsab1 2811. UPDATE / TODO: nfsab1 2811 does not use ax-13 2392 either anymore; bj-nfsab1 34167 is shorter than nfsab1 2811 but uses ax-12 2179. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
𝑥 𝑦 ∈ {𝑥𝜑}

Theorembj-dtru 34168* Remove dependency on ax-13 2392 from dtru 5259. (Contributed by BJ, 31-May-2019.)

TODO: This predates the removal of ax-13 2392 in dtru 5259. But actually, sn-dtru 39285 is better than either, so move it to Main with sn-el 39284 (and determine whether bj-dtru 34168 should be kept as ALT or deleted).

(Proof modification is discouraged.) (New usage is discouraged.)

¬ ∀𝑥 𝑥 = 𝑦

Theorembj-dtrucor2v 34169* Version of dtrucor2 5261 with a disjoint variable condition, which does not require ax-13 2392 (nor ax-4 1811, ax-5 1912, ax-7 2016, ax-12 2179). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦𝑥𝑦)       (𝜑 ∧ ¬ 𝜑)

20.15.4.13  Distinct var metavariables

The closed formula 𝑥𝑦𝑥 = 𝑦 approximately means that the var metavariables 𝑥 and 𝑦 represent the same variable vi. In a domain with at most one object, however, this formula is always true, hence the "approximately" in the previous sentence.

Theorembj-hbaeb2 34170 Biconditional version of a form of hbae 2455 with commuted quantifiers, not requiring ax-11 2162. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑧 𝑥 = 𝑦)

Theorembj-hbaeb 34171 Biconditional version of hbae 2455. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧𝑥 𝑥 = 𝑦)

Theorembj-hbnaeb 34172 Biconditional version of hbnae 2456 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theorembj-dvv 34173 A special instance of bj-hbaeb2 34170. A lemma for distinct var metavariables. Note that the right-hand side is a closed formula (a sentence). (Contributed by BJ, 6-Oct-2018.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑦 𝑥 = 𝑦)

20.15.4.14  Around ~ equsal

As a rule of thumb, if a theorem of the form (𝜑𝜓) ⇒ (𝜒𝜃) is in the database, and the "more precise" theorems (𝜑𝜓) ⇒ (𝜒𝜃) and (𝜓𝜑) ⇒ (𝜃𝜒) also hold (see bj-bisym 33951), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2442 (and equsalh 2444 and equsexh 2445). Even if only one of these two theorems holds, it should be added to the database.

Theorembj-equsal1t 34174 Duplication of wl-equsal1t 34858, with shorter proof. If one imposes a disjoint variable condition on x,y , then one can use alequexv 2008 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 34859 is also interesting. (Contributed by BJ, 6-Oct-2018.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))

Theorembj-equsal1ti 34175 Inference associated with bj-equsal1t 34174. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)

Theorembj-equsal1 34176 One direction of equsal 2441. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) → 𝜓)

Theorembj-equsal2 34177 One direction of equsal 2441. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓))

Theorembj-equsal 34178 Shorter proof of equsal 2441. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2441, but "min */exc equsal" is ok. (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

20.15.4.15  Some Principia Mathematica proofs

References are made to the second edition (1927, reprinted 1963) of Principia Mathematica, Vol. 1. Theorems are referred to in the form "PM*xx.xx".

Theoremstdpc5t 34179 Closed form of stdpc5 2210. (Possible to place it before 19.21t 2208 and use it to prove 19.21t 2208). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Theorembj-stdpc5 34180 More direct proof of stdpc5 2210. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theorem2stdpc5 34181 A double stdpc5 2210 (one direction of PM*11.3). See also 2stdpc4 2076 and 19.21vv 40940. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) → (𝜑 → ∀𝑥𝑦𝜓))

Theorembj-19.21t0 34182 Proof of 19.21t 2208 from stdpc5t 34179. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theoremexlimii 34183 Inference associated with exlimi 2219. Inferring a theorem when it is implied by an antecedent which may be true. (Contributed by BJ, 15-Sep-2018.)
𝑥𝜓    &   (𝜑𝜓)    &   𝑥𝜑       𝜓

Theoremax11-pm 34184 Proof of ax-11 2162 similar to PM's proof of alcom 2164 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 34188. Axiom ax-11 2162 is used in the proof only through nfa2 2178. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremax6er 34185 Commuted form of ax6e 2403. (Could be placed right after ax6e 2403). (Contributed by BJ, 15-Sep-2018.)
𝑥 𝑦 = 𝑥

Theoremexlimiieq1 34186 Inferring a theorem when it is implied by an equality which may be true. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑    &   (𝑥 = 𝑦𝜑)       𝜑

Theoremexlimiieq2 34187 Inferring a theorem when it is implied by an equality which may be true. (Contributed by BJ, 15-Sep-2018.) (Revised by BJ, 30-Sep-2018.)
𝑦𝜑    &   (𝑥 = 𝑦𝜑)       𝜑

Theoremax11-pm2 34188* Proof of ax-11 2162 from the standard axioms of predicate calculus, similar to PM's proof of alcom 2164 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 2162 is used in the proof only through nfal 2344, nfsb 2567, sbal 2167, sb8 2561. See also ax11-pm 34184. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

20.15.4.16  Alternate definition of substitution

Theorembj-sbsb 34189 Biconditional showing two possible (dual) definitions of substitution df-sb 2071 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
(((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜑)))

Theorembj-dfsb2 34190 Alternate (dual) definition of substitution df-sb 2071 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
([𝑦 / 𝑥]𝜑 ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜑)))

20.15.4.17  Lemmas for substitution

Theorembj-sbf3 34191 Substitution has no effect on a bound variable (existential quantifier case); see sbf2 2274. (Contributed by BJ, 2-May-2019.)
([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)

Theorembj-sbf4 34192 Substitution has no effect on a bound variable (nonfreeness case); see sbf2 2274. (Contributed by BJ, 2-May-2019.)
([𝑦 / 𝑥]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜑)

Theorembj-sbnf 34193* Move nonfree predicate in and out of substitution; see sbal 2167 and sbex 2290. (Contributed by BJ, 2-May-2019.)
([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)

20.15.4.18  Existential uniqueness

Theorembj-eu3f 34194* Version of eu3v 2656 where the disjoint variable condition is replaced with a nonfreeness hypothesis. This is a "backup" of a theorem that used to be in the main part with label "eu3" and was deprecated in favor of eu3v 2656. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)
𝑦𝜑       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

20.15.4.19  First-order logic: miscellaneous

Miscellaneous theorems of first-order logic.

Theorembj-sblem1 34195* Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜒)))

Theorembj-sblem2 34196* Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
(∀𝑥(𝜑 → (𝜒𝜓)) → ((∃𝑥𝜑𝜒) → ∀𝑥(𝜑𝜓)))

Theorembj-sblem 34197* Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜒)))

Theorembj-sbievw1 34198* Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑𝜓))

Theorembj-sbievw2 34199* Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
([𝑦 / 𝑥](𝜓𝜑) → (𝜓 → [𝑦 / 𝑥]𝜑))

Theorembj-sbievw 34200* Lemma for substitution. Closed form of equsalvw 2011 and sbievw 2104. (Contributed by BJ, 23-Jul-2023.)
([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑𝜓))

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