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Theorem List for Metamath Proof Explorer - 32701-32800   *Has distinct variable group(s)
TypeLabelDescription
Statement

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

It is known that ax-13 2245 is logically redundant (see ax13w 2012 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2245 from every theorem in set.mm which is totally unbundled (i.e., has dv conditions on all setvar variables). Indeed, start with the existing proof, and replace any occurrence of ax-13 2245 with ax13w 2012.

This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2245 (and using ax6v 1888 / ax6ev 1889 instead of ax-6 1887 / ax6e 2249, as is currently done).

One reason to be optimistic is that the first few utility theorems using ax-13 2245 (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 2245, labeled bj-xxxv (we follow the proof of xxx but use ax6v 1888 and ax6ev 1889 instead of ax-6 1887 and ax6e 2249, and ax-5 1838 instead of ax13v 2246; 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 2245, so as to remove dependencies on ax-13 2245 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 2033, typically by replacing a non-free hypothesis with a dv condition (see bj-cbv3v2 32711 and following theorems).

Theorembj-axc10v 32701* Version of axc10 2251 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Theorembj-spimtv 32702* Version of spimt 2252 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))

Theorembj-spimedv 32703* Version of spimed 2254 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝜒 → Ⅎ𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (𝜑 → ∃𝑥𝜓))

Theorembj-spimev 32704* Version of spime 2255 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theorembj-spimvv 32705* Version of spimv 2256 and spimv1 2114 with a dv condition, which does not require ax-13 2245. UPDATE: this is spimvw 1926. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theorembj-spimevv 32706* Version of spimev 2258 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theorembj-spvv 32707* Version of spv 2259 with a dv condition, which does not require ax-7 1934, ax-12 2046, ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theorembj-speiv 32708* Version of spei 2260 with a dv condition, which does not require ax-13 2245 (neither ax-7 1934 nor ax-12 2046). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑

Theorembj-chvarv 32709* Version of chvar 2261 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theorembj-chvarvv 32710* Version of chvarv 2262 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theorembj-cbv3v2 32711* Version of cbv3 2264 with two dv conditions, which does not require ax-11 2033 nor ax-13 2245. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theorembj-cbv3hv2 32712* Version of cbv3h 2265 with two dv conditions, which does not require ax-11 2033 nor ax-13 2245. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theorembj-cbv1v 32713* Version of cbv1 2266 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theorembj-cbv1hv 32714* Version of cbv1h 2267 with a dv condition, which does not require ax-13 2245. (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 32715* Version of cbv2h 2268 with a dv condition, which does not require ax-13 2245. (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 32716* Version of cbv2 2269 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theorembj-cbvalvv 32717* Version of cbvalv 2272 with a dv condition, which does not require ax-13 2245. UPDATE: this is cbvalvw 1968 (which is proved with fewer axioms). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theorembj-cbvexvv 32718* Version of cbvexv 2274 with a dv condition, which does not require ax-13 2245. UPDATE: this is cbvexvw 1969 (which is proved with fewer axioms). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theorembj-cbvaldv 32719* Version of cbvald 2276 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theorembj-cbvexdv 32720* Version of cbvexd 2277 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theorembj-cbval2v 32721* Version of cbval2 2278 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theorembj-cbvex2v 32722* Version of cbvex2 2279 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theorembj-cbval2vv 32723* Version of cbval2v 2284 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theorembj-cbvex2vv 32724* Version of cbvex2v 2286 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theorembj-cbvaldvav 32725* Version of cbvaldva 2280 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theorembj-cbvexdvav 32726* Version of cbvexdva 2282 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theorembj-cbvex4vv 32727* Version of cbvex4v 2288 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))    &   ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))       (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)

Theorembj-equsalhv 32728* Version of equsalh 2293 with a dv condition, which does not require ax-13 2245. Remark: this is the same as equsalhw 2122.

Remarks: equsexvw 1931 has been moved to Main; the theorem ax13lem2 2295 has a dv version which is a simple consequence of ax5e 1840; the theorems nfeqf2 2296, dveeq2 2297, nfeqf1 2298, dveeq1 2299, nfeqf 2300, axc9 2301, ax13 2248, have dv versions which are simple consequences of ax-5 1838. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)

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

Theorembj-axc11nv 32729* Version of axc11n 2306 with a dv condition; instance of aevlem 1980. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theorembj-aecomsv 32730* Version of aecoms 2311 with a dv condition, provable from Tarski's FOL. The corresponding version of naecoms 2312 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV(x, y) is true when the universe has at least two objects (see bj-dtru 32781). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theorembj-axc11v 32731* Version of axc11 2313 with a dv condition, which does not require ax-13 2245 nor ax-10 2018. Remark: the following theorems (hbae 2314, nfae 2315, hbnae 2316, nfnae 2317, hbnaes 2318) would need to be totally unbundled to be proved without ax-13 2245, hence would be simple consequences of ax-5 1838 or nfv 1842. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theorembj-dral1v 32732* Version of dral1 2324 with a dv condition, which does not require ax-13 2245. Remark: the corresponding versions for dral2 2323 and drex2 2327 are instances of albidv 1848 and exbidv 1849 respectively. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theorembj-drex1v 32733* Version of drex1 2326 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Theorembj-drnf1v 32734* Version of drnf1 2328 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Theorembj-drnf2v 32735* Version of drnf2 2329 with a dv condition, which does not require ax-13 2245. Could be labeled "nfbidv". Note that the version of axc15 2302 with a dv condition is actually ax12v2 2048 (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 32736* Version of equs45f 2349 with a dv condition, which does not require ax-13 2245. Note that the version of equs5 2350 with a dv condition is actually sb56 2149 (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
𝑦𝜑       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-sb2v 32737* Version of sb2 2351 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Theorembj-stdpc4v 32738* Version of stdpc4 2352 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Theorembj-2stdpc4v 32739* Version of 2stdpc4 2353 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Theorembj-sb3v 32740* Version of sb3 2354 with a dv condition, which does not require ax-13 2245. This allows to remove ax-13 2245 from sb5 2429 (see bj-sb5 32752). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Theorembj-sb4v 32741* Version of sb4 2355 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 23-Jun-2019.) Together with bj-sb2v 32737, this allosw to remove ax-13 2245 from sb6 2428 (see bj-sb6 32751). Note that this subsumes the version of sb4b 2357 with a dv condition. (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-hbs1 32742* Version of hbsb2 2358 with a dv condition, which does not require ax-13 2245, and removal of ax-13 2245 from hbs1 2435. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-nfs1v 32743* Version of nfsb2 2359 with a dv condition, which does not require ax-13 2245, and removal of ax-13 2245 from nfs1v 2436. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
𝑥[𝑦 / 𝑥]𝜑

Theorembj-hbsb2av 32744* Version of hbsb2a 2360 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-hbsb3v 32745* Version of hbsb3 2363 with a dv condition, which does not require ax-13 2245. (Remark: the unbundled version of nfs1 2364 is given by bj-nfs1v 32743.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-equsb1v 32746* Version of equsb1 2367 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
[𝑦 / 𝑥]𝑥 = 𝑦

Theorembj-sbftv 32747* Version of sbft 2378 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))

Theorembj-sbfv 32748* Version of sbf 2379 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥𝜑       ([𝑦 / 𝑥]𝜑𝜑)

Theorembj-sbfvv 32749* Version of sbf 2379 with two dv conditions, which does not require ax-10 2018 nor ax-13 2245. (Contributed by BJ, 1-May-2021.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑𝜑)

Theorembj-sbtv 32750* Version of sbt 2418 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝜑       [𝑦 / 𝑥]𝜑

Theorembj-sb6 32751* Remove dependency on ax-13 2245 from sb6 2428. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-sb5 32752* Remove dependency on ax-13 2245 from sb5 2429. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theorembj-axext3 32753* Remove dependency on ax-13 2245 from axext3 2603. (Contributed by BJ, 12-Jul-2019.) (Proof modification is discouraged.)
(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)

Theorembj-axext4 32754* Remove dependency on ax-13 2245 from axext4 2605. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))

Theorembj-hbab1 32755* Remove dependency on ax-13 2245 from hbab1 2610. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})

Theorembj-nfsab1 32756* Remove dependency on ax-13 2245 from nfsab1 2611. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
𝑥 𝑦 ∈ {𝑥𝜑}

Theorembj-abeq2 32757* Remove dependency on ax-13 2245 from abeq2 2731. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theorembj-abeq1 32758* Remove dependency on ax-13 2245 from abeq1 2732. Remark: the theorems abeq2i 2734, abeq1i 2735, abeq2d 2733 do not use ax-11 2033 or ax-13 2245. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theorembj-abbi 32759 Remove dependency on ax-13 2245 from abbi 2736. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})

Theorembj-abbi2i 32760* Remove dependency on ax-13 2245 from abbi2i 2737. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}

Theorembj-abbii 32761 Remove dependency on ax-13 2245 from abbii 2738. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}

Theorembj-abbid 32762 Remove dependency on ax-13 2245 from abbid 2739. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Theorembj-abbidv 32763* Remove dependency on ax-13 2245 from abbidv 2740. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Theorembj-abbi2dv 32764* Remove dependency on ax-13 2245 from abbi2dv 2741. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})

Theorembj-abbi1dv 32765* Remove dependency on ax-13 2245 from abbi1dv 2742. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} = 𝐴)

Theorembj-abid2 32766* Remove dependency on ax-13 2245 from abid2 2744. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
{𝑥𝑥𝐴} = 𝐴

Theorembj-clabel 32767* Remove dependency on ax-13 2245 from clabel 2748 (note the absence of DV conditions among variables in the LHS). (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))

Theorembj-sbab 32768* Remove dependency on ax-13 2245 from sbab 2749 (note the absence of DV conditions among variables in the LHS). (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧𝐴})

Theorembj-nfab1 32769 Remove dependency on ax-13 2245 from nfab1 2765 (note the absence of DV conditions). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥{𝑥𝜑}

Theorembj-vjust 32770 Remove dependency on ax-13 2245 from vjust 3199 (note the absence of DV conditions). Soundness justification theorem for df-v 3200. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
{𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}

Theorembj-cdeqab 32771* Remove dependency on ax-13 2245 from cdeqab 3423. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})

Theorembj-axrep1 32772* Remove dependency on ax-13 2245 from axrep1 4770. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))

Theorembj-axrep2 32773* Remove dependency on ax-13 2245 from axrep2 4771. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))

Theorembj-axrep3 32774* Remove dependency on ax-13 2245 from axrep3 4772. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))

Theorembj-axrep4 32775* Remove dependency on ax-13 2245 from axrep4 4773. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑧𝜑       (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))

Theorembj-axrep5 32776* Remove dependency on ax-13 2245 from axrep5 4774. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑧𝜑       (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))

Theorembj-axsep 32777* Remove dependency on ax-13 2245 from axsep 4778. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theorembj-nalset 32778* Remove dependency on ax-13 2245 from nalset 4793. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
¬ ∃𝑥𝑦 𝑦𝑥

Theorembj-zfpow 32779* Remove dependency on ax-13 2245 from zfpow 4842. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)

Theorembj-el 32780* Remove dependency on ax-13 2245 from el 4845. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑦 𝑥𝑦

Theorembj-dtru 32781* Remove dependency on ax-13 2245 from dtru 4855. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
¬ ∀𝑥 𝑥 = 𝑦

Theorembj-axc16b 32782* Remove dependency on ax-13 2245 from axc16b 4856. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theorembj-eunex 32783 Remove dependency on ax-13 2245 from eunex 4857. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Theorembj-dtrucor 32784* Remove dependency on ax-13 2245 from dtrucor 4898. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥 = 𝑦       𝑥𝑦

Theorembj-dtrucor2v 32785* Version of dtrucor2 4899 with a dv condition, which does not require ax-13 2245 (nor ax-4 1736, ax-5 1838, ax-7 1934, ax-12 2046). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦𝑥𝑦)       (𝜑 ∧ ¬ 𝜑)

Theorembj-dvdemo1 32786* Remove dependency on ax-13 2245 from dvdemo1 4900 (this removal is noteworthy since dvdemo1 4900 and dvdemo2 4901 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑦𝑧𝑥)

Theorembj-dvdemo2 32787* Remove dependency on ax-13 2245 from dvdemo2 4901 (this removal is noteworthy since dvdemo1 4900 and dvdemo2 4901 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑦𝑧𝑥)

20.14.4.12  Strengthenings of theorems of the main part

Typically, these are biconditional versions of theorems in the main part which are formulated as implications. They could be added after said implication, or sometimes replace it (by "inlining" it).

This could also be done for hba1 2150, hbe1 2020, hbn1 2019, modal-5 2031.

Theorembj-sb3b 32788 Simplified definition of substitution when variables are distinct. This is to sb3 2354 what sb4b 2357 is to sb4 2355. Actually, one may keep only bj-sb3b 32788 and sb4b 2357 in the database, renaming them sb3 and sb4. (Contributed by BJ, 6-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))

20.14.4.13  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 32789 Biconditional version of a form of hbae 2314 with commuted quantifiers, not requiring ax-11 2033. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑧 𝑥 = 𝑦)

Theorembj-hbaeb 32790 Biconditional version of hbae 2314. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧𝑥 𝑥 = 𝑦)

Theorembj-hbnaeb 32791 Biconditional version of hbnae 2316 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theorembj-dvv 32792 A special instance of bj-hbaeb2 32789. 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)

20.14.4.14  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 32559), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2291 (and equsalh 2293 and equsexh 2294). Even if only one of these two theorems holds, it should be added to the database.

Theorembj-equsal1t 32793 Duplication of wl-equsal1t 33307, with shorter proof. If one imposes a DV condition on x,y , then one can use bj-alequexv 32639 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 33308 is also interesting. (Contributed by BJ, 6-Oct-2018.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))

Theorembj-equsal1ti 32794 Inference associated with bj-equsal1t 32793. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)

Theorembj-equsal1 32795 One direction of equsal 2290. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) → 𝜓)

Theorembj-equsal2 32796 One direction of equsal 2290. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓))

Theorembj-equsal 32797 Shorter proof of equsal 2290. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2290, but "min */exc equsal" is ok. (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

20.14.4.15  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 32798 Closed form of stdpc5 2075. (Possible to place it before 19.21t 2072 and use it to prove 19.21t 2072). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Theorembj-stdpc5 32799 More direct proof of stdpc5 2075. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theorem2stdpc5 32800 A double stdpc5 2075 (one direction of PM*11.3). See also 2stdpc4 2353 and 19.21vv 38401. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) → (𝜑 → ∀𝑥𝑦𝜓))

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