Home Metamath Proof ExplorerTheorem List (p. 334 of 437) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28347) Hilbert Space Explorer (28348-29872) Users' Mathboxes (29873-43661)

Theorem List for Metamath Proof Explorer - 33301-33400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-hbsb3 33301 Shorter proof of hbsb3 2440. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-nfs1t 33302 A theorem close to a closed form of nfs1 2441. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theorembj-nfs1t2 33303 A theorem close to a closed form of nfs1 2441. (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theorembj-nfs1 33304 Shorter proof of nfs1 2441 (three essential steps instead of four). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
𝑦𝜑       𝑥[𝑦 / 𝑥]𝜑

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

It is known that ax-13 2334 is logically redundant (see ax13w 2130 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2334 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 2334 with ax13w 2130.

This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2334 (and using ax6v 2022 / ax6ev 2023 instead of ax-6 2021 / ax6e 2347, as is currently done).

One reason to be optimistic is that the first few utility theorems using ax-13 2334 (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 2334, labeled bj-xxxv (we follow the proof of xxx but use ax6v 2022 and ax6ev 2023 instead of ax-6 2021 and ax6e 2347, and ax-5 1953 instead of ax13v 2335; 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 2334, so as to remove dependencies on ax-13 2334 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 2150, typically by replacing a non-free hypothesis with a disjoint variable condition (see cbv3v2 2240 and following theorems).

Theorembj-axc10v 33305* Version of axc10 2349 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Theorembj-spimtv 33306* Version of spimt 2350 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))

Theorembj-spimedv 33307* Version of spimed 2353 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝜒 → Ⅎ𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (𝜑 → ∃𝑥𝜓))

Theorembj-spimev 33308* Version of spime 2354 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theorembj-spimvv 33309* Version of spimv 2355 and spimv1 2239 with a disjoint variable condition, which does not require ax-13 2334. UPDATE: this is spimvw 2045 (but different proof). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theorembj-spimevv 33310* Version of spimev 2357 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theorembj-spvv 33311* Version of spv 2358 with a disjoint variable condition, which does not require ax-7 2055, ax-12 2163, ax-13 2334. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theorembj-speiv 33312* Version of spei 2359 with a disjoint variable condition, which does not require ax-13 2334 (neither ax-7 2055 nor ax-12 2163). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑

Theorembj-chvarv 33313* Version of chvar 2360 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theorembj-chvarvv 33314* Version of chvarv 2361 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theorembj-cbv3hv2 33315* Version of cbv3h 2363 with two disjoint variable conditions, which does not require ax-11 2150 nor ax-13 2334. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theorembj-cbv1v 33316* Version of cbv1 2364 with a disjoint variable condition, which does not require ax-13 2334. (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 33317* Version of cbv1h 2365 with a disjoint variable condition, which does not require ax-13 2334. (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 33318* Version of cbv2h 2366 with a disjoint variable condition, which does not require ax-13 2334. (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 33319* Version of cbv2 2367 with a disjoint variable condition, which does not require ax-13 2334. (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 33320* Version of cbvald 2372 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theorembj-cbvexdv 33321* Version of cbvexd 2373 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theorembj-cbval2v 33322* Version of cbval2 2374 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theorembj-cbvex2v 33323* Version of cbvex2 2375 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theorembj-cbval2vv 33324* Version of cbval2v 2378 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theorembj-cbvex2vv 33325* Version of cbvex2v 2379 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theorembj-cbvaldvav 33326* Version of cbvaldva 2376 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theorembj-cbvexdvav 33327* Version of cbvexdva 2377 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theorembj-cbvex4vv 33328* Version of cbvex4v 2380 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))    &   ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))       (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)

Theorembj-equsalhv 33329* Version of equsalh 2385 with a disjoint variable condition, which does not require ax-13 2334. Remark: this is the same as equsalhw 2265 (TODO: delete after moving the following paragraph somewhere).

Remarks: equsexvw 2052 has been moved to Main; the theorem ax13lem2 2338 has a dv version which is a simple consequence of ax5e 1955; the theorems nfeqf2 2339, dveeq2 2342, nfeqf1 2343, dveeq1 2344, nfeqf 2345, axc9 2346, ax13 2337, have dv versions which are simple consequences of ax-5 1953. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)

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

Theorembj-axc11nv 33330* Version of axc11n 2392 with a disjoint variable condition; instance of aevlem 2098 (TODO: delete after checking surrounding theorems). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theorembj-aecomsv 33331* Version of aecoms 2394 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2395 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV(x,y) is true when the universe has at least two objects (see bj-dtru 33373). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theorembj-axc11v 33332* Version of axc11 2396 with a disjoint variable condition, which does not require ax-13 2334 nor ax-10 2135. Remark: the following theorems (hbae 2397, nfae 2398, hbnae 2399, nfnae 2400, hbnaes 2401) would need to be totally unbundled to be proved without ax-13 2334, hence would be simple consequences of ax-5 1953 or nfv 1957. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theorembj-dral1v 33333* Version of dral1 2405 with a disjoint variable condition, which does not require ax-13 2334. Remark: the corresponding versions for dral2 2404 and drex2 2408 are instances of albidv 1963 and exbidv 1964 respectively. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theorembj-drex1v 33334* Version of drex1 2407 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Theorembj-drnf1v 33335* Version of drnf1 2409 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Theorembj-drnf2v 33336* Version of drnf2 2410 with a disjoint variable condition, which does not require ax-10 2135, ax-11 2150, ax-12 2163, ax-13 2334. Instance of nfbidv 1965. Note that the version of axc15 2387 with a disjoint variable condition is actually ax12v2 2165 (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 33337* Version of equs45f 2425 with a disjoint variable condition, which does not require ax-13 2334. Note that the version of equs5 2426 with a disjoint variable condition is actually sb56 2248 (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
𝑦𝜑       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theorembj-2stdpc4v 33338* Version of 2stdpc4 2429 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Theorembj-sb3v 33339* Version of sb3 2430 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Theorembj-hbs1 33340* Version of hbsb2 2435 with a disjoint variable condition, which does not require ax-13 2334, and removal of ax-13 2334 from hbs1 2255. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-nfs1v 33341* Version of nfsb2 2436 with a disjoint variable condition, which does not require ax-13 2334, and removal of ax-13 2334 from nfs1v 2254. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
𝑥[𝑦 / 𝑥]𝜑

Theorembj-hbsb2av 33342* Version of hbsb2a 2437 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-hbsb3v 33343* Version of hbsb3 2440 with a disjoint variable condition, which does not require ax-13 2334. (Remark: the unbundled version of nfs1 2441 is given by bj-nfs1v 33341.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theorembj-sbfvv 33344* Version of sbf 2456 with two disjoint variable conditions, which does not require ax-10 2135 nor ax-13 2334. (Contributed by BJ, 1-May-2021.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑𝜑)

Theorembj-sbtv 33345* Version of sbt 2496 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝜑       [𝑦 / 𝑥]𝜑

Theorembj-axext3 33346* Remove dependency on ax-13 2334 from axext3 2756. (Contributed by BJ, 12-Jul-2019.) (Proof modification is discouraged.)
(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)

Theorembj-axext4 33347* Remove dependency on ax-13 2334 from axext4 2758. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))

Theorembj-hbab1 33348* Remove dependency on ax-13 2334 from hbab1 2766. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})

Theorembj-nfsab1 33349* Remove dependency on ax-13 2334 from nfsab1 2767. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
𝑥 𝑦 ∈ {𝑥𝜑}

Theorembj-abeq2 33350* Remove dependency on ax-13 2334 from abeq2 2892. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theorembj-abeq1 33351* Remove dependency on ax-13 2334 from abeq1 2893. Remark: the theorems abeq2i 2895, abeq1i 2896, abeq2d 2894 do not use ax-11 2150 or ax-13 2334. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theorembj-abbi 33352 Remove dependency on ax-13 2334 from abbi 2902. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})

Theorembj-abbi2i 33353* Remove dependency on ax-13 2334 from abbi2i 2900. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}

Theorembj-abbii 33354 Remove dependency on ax-13 2334 from abbii 2908. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}

Theorembj-abbid 33355 Remove dependency on ax-13 2334 from abbid 2905. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Theorembj-abbidv 33356* Remove dependency on ax-13 2334 from abbidv 2906. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Theorembj-abbi2dv 33357* Remove dependency on ax-13 2334 from abbi2dv 2897. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})

Theorembj-abbi1dv 33358* Remove dependency on ax-13 2334 from abbi1dv 2899. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} = 𝐴)

Theorembj-abid2 33359* Remove dependency on ax-13 2334 from abid2 2912. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
{𝑥𝑥𝐴} = 𝐴

Theorembj-clabel 33360* Remove dependency on ax-13 2334 from clabel 2917 (note the absence of disjoint variable conditions among variables in the LHS). (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))

Theorembj-sbab 33361* Remove dependency on ax-13 2334 from sbab 2918 (note the absence of disjoint variable conditions among variables in the LHS). (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧𝐴})

Theorembj-nfab1 33362 Remove dependency on ax-13 2334 from nfab1 2936 (note the absence of disjoint variable conditions). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥{𝑥𝜑}

Theorembj-vjust 33363 Remove dependency on ax-13 2334 from vjust 3399 (note the absence of disjoint variable conditions). Soundness justification theorem for df-v 3400. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
{𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}

Theorembj-cdeqab 33364* Remove dependency on ax-13 2334 from cdeqab 3642. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})

Theorembj-axrep1 33365* Remove dependency on ax-13 2334 from axrep1 5007. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))

Theorembj-axrep2 33366* Remove dependency on ax-13 2334 from axrep2 5009. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))

Theorembj-axrep3 33367* Remove dependency on ax-13 2334 from axrep3 5010. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))

Theorembj-axrep4 33368* Remove dependency on ax-13 2334 from axrep4 5011. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑧𝜑       (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))

Theorembj-axrep5 33369* Remove dependency on ax-13 2334 from axrep5 5012. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑧𝜑       (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))

Theorembj-axsep 33370* Remove dependency on ax-13 2334 from axsep 5016. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theorembj-nalset 33371* Remove dependency on ax-12 2163 and ax-13 2334 (and df-nf 1828) from nalset 5032. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
¬ ∃𝑥𝑦 𝑦𝑥

Theorembj-el 33372* Remove dependency on ax-13 2334 from el 5081. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
𝑦 𝑥𝑦

Theorembj-dtru 33373* Remove dependency on ax-13 2334 from dtru 5082. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
¬ ∀𝑥 𝑥 = 𝑦

Theorembj-axc16b 33374* Remove dependency on ax-13 2334 from axc16b 5100. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theorembj-eunex 33375 Remove dependency on ax-13 2334 from eunex 5101. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Theorembj-dtrucor 33376* Remove dependency on ax-13 2334 from dtrucor 5083. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥 = 𝑦       𝑥𝑦

Theorembj-dtrucor2v 33377* Version of dtrucor2 5084 with a disjoint variable condition, which does not require ax-13 2334 (nor ax-4 1853, ax-5 1953, ax-7 2055, ax-12 2163). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦𝑥𝑦)       (𝜑 ∧ ¬ 𝜑)

Theorembj-dvdemo1 33378* Remove dependency on ax-13 2334 from dvdemo1 5085 (this removal is noteworthy since dvdemo1 5085 and dvdemo2 5086 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑦𝑧𝑥)

Theorembj-dvdemo2 33379* Remove dependency on ax-13 2334 from dvdemo2 5086 (this removal is noteworthy since dvdemo1 5085 and dvdemo2 5086 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
𝑥(𝑥 = 𝑦𝑧𝑥)

20.14.4.12  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 33380 Biconditional version of a form of hbae 2397 with commuted quantifiers, not requiring ax-11 2150. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑧 𝑥 = 𝑦)

Theorembj-hbaeb 33381 Biconditional version of hbae 2397. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧𝑥 𝑥 = 𝑦)

Theorembj-hbnaeb 33382 Biconditional version of hbnae 2399 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theorembj-dvv 33383 A special instance of bj-hbaeb2 33380. 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)

20.14.4.13  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 33154), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2383 (and equsalh 2385 and equsexh 2386). Even if only one of these two theorems holds, it should be added to the database.

Theorembj-equsal1t 33384 Duplication of wl-equsal1t 33921, with shorter proof. If one imposes a disjoint variable condition on x,y , then one can use bj-alequexv 33244 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 33922 is also interesting. (Contributed by BJ, 6-Oct-2018.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))

Theorembj-equsal1ti 33385 Inference associated with bj-equsal1t 33384. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)

Theorembj-equsal1 33386 One direction of equsal 2382. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) → 𝜓)

Theorembj-equsal2 33387 One direction of equsal 2382. (Contributed by BJ, 30-Sep-2018.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓))

Theorembj-equsal 33388 Shorter proof of equsal 2382. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2382, but "min */exc equsal" is ok. (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

20.14.4.14  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 33389 Closed form of stdpc5 2193. (Possible to place it before 19.21t 2191 and use it to prove 19.21t 2191). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Theorembj-stdpc5 33390 More direct proof of stdpc5 2193. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theorem2stdpc5 33391 A double stdpc5 2193 (one direction of PM*11.3). See also 2stdpc4 2429 and 19.21vv 39535. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) → (𝜑 → ∀𝑥𝑦𝜓))

Theorembj-19.21t 33392 Proof of 19.21t 2191 from stdpc5t 33389. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theoremexlimii 33393 Inference associated with exlimi 2203. 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 33394 Proof of ax-11 2150 similar to PM's proof of alcom 2152 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 33398. Axiom ax-11 2150 is used in the proof only through nfa2 2162. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremax6er 33395 Commuted form of ax6e 2347. (Could be placed right after ax6e 2347). (Contributed by BJ, 15-Sep-2018.)
𝑥 𝑦 = 𝑥

Theoremexlimiieq1 33396 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 33397 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 33398* Proof of ax-11 2150 from the standard axioms of predicate calculus, similar to PM's proof of alcom 2152 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 2150 is used in the proof only through nfal 2299, nfsb 2520, sbal 2542, sb8 2501. See also ax11-pm 33394. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

20.14.4.15  Alternate definition of substitution

Theorembj-sbsb 33399 Biconditional showing two possible (dual) definitions of substitution df-sb 2012 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
(((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜑)))

Theorembj-dfsb2 33400 Alternate (dual) definition of substitution df-sb 2012 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
([𝑦 / 𝑥]𝜑 ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜑)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43661
 Copyright terms: Public domain < Previous  Next >