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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-19.41t 34001 | Closed form of 19.41 2228 from the same axioms as 19.41v 1941. The same is doable with 19.27 2220, 19.28 2221, 19.31 2227, 19.32 2226, 19.44 2230, 19.45 2231. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))) | ||
Theorem | bj-sbft 34002 | Version of sbft 2261 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.) |
⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | bj-axc10 34003 | Alternate (shorter) proof of axc10 2396. One can prove a version with DV (𝑥, 𝑦) without ax-13 2383, by using ax6ev 1963 instead of ax6e 2394. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Theorem | bj-alequex 34004 | A fol lemma. See alequexv 1998 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2397 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | ||
Theorem | bj-spimt2 34005 | A step in the proof of spimt 2397. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓))) | ||
Theorem | bj-cbv3ta 34006 | Closed form of cbv3 2409. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-cbv3tb 34007 | Closed form of cbv3 2409. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-hbsb3t 34008 | A theorem close to a closed form of hbsb3 2522. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)) | ||
Theorem | bj-hbsb3 34009 | Shorter proof of hbsb3 2522. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1t 34010 | A theorem close to a closed form of nfs1 2523. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1t2 34011 | A theorem close to a closed form of nfs1 2523. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1 34012 | Shorter proof of nfs1 2523 (three essential steps instead of four). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | ||
It is known that ax-13 2383 is logically redundant (see ax13w 2131 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2383 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 2383 with ax13w 2131. This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2383 (and using ax6v 1962 / ax6ev 1963 instead of ax-6 1961 / ax6e 2394, as is currently done). One reason to be optimistic is that the first few utility theorems using ax-13 2383 (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 2383, labeled bj-xxxv (we follow the proof of xxx but use ax6v 1962 and ax6ev 1963 instead of ax-6 1961 and ax6e 2394, and ax-5 1902 instead of ax13v 2384; 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 2383, so as to remove dependencies on ax-13 2383 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 2151, typically by replacing a nonfree hypothesis with a disjoint variable condition (see cbv3v2 2234 and following theorems). | ||
Theorem | bj-axc10v 34013* | Version of axc10 2396 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Theorem | bj-spimtv 34014* | Version of spimt 2397 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-cbv3hv2 34015* | Version of cbv3h 2418 with two disjoint variable conditions, which does not require ax-11 2151 nor ax-13 2383. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | bj-cbv1hv 34016* | Version of cbv1h 2419 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) | ||
Theorem | bj-cbv2hv 34017* | Version of cbv2h 2420 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | bj-cbv2v 34018* | Version of cbv2 2417 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | bj-cbvaldv 34019* | Version of cbvald 2422 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | bj-cbvexdv 34020* | Version of cbvexd 2423 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
Theorem | bj-cbval2vv 34021* | Version of cbval2vv 2429 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) | ||
Theorem | bj-cbvex2vv 34022* | Version of cbvex2vv 2430 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) | ||
Theorem | bj-cbvaldvav 34023* | Version of cbvaldva 2424 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | bj-cbvexdvav 34024* | Version of cbvexdva 2425 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
Theorem | bj-cbvex4vv 34025* | Version of cbvex4v 2431 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) & ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) | ||
Theorem | bj-equsalhv 34026* |
Version of equsalh 2436 with a disjoint variable condition, which
does not
require ax-13 2383. Remark: this is the same as equsalhw 2291. TODO:
delete after moving the following paragraph somewhere.
Remarks: equsexvw 2002 has been moved to Main; the theorem ax13lem2 2387 has a dv version which is a simple consequence of ax5e 1904; the theorems nfeqf2 2388, dveeq2 2389, nfeqf1 2390, dveeq1 2391, nfeqf 2392, axc9 2393, ax13 2386, have dv versions which are simple consequences of ax-5 1902. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | bj-axc11nv 34027* | Version of axc11n 2443 with a disjoint variable condition; instance of aevlem 2051. 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) | ||
Theorem | bj-aecomsv 34028* | Version of aecoms 2445 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2446 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5263). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | bj-axc11v 34029* | Version of axc11 2447 with a disjoint variable condition, which does not require ax-13 2383 nor ax-10 2136. Remark: the following theorems (hbae 2448, nfae 2450, hbnae 2449, nfnae 2451, hbnaes 2452) would need to be totally unbundled to be proved without ax-13 2383, hence would be simple consequences of ax-5 1902 or nfv 1906. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Theorem | bj-drnf2v 34030* | Version of drnf2 2461 with a disjoint variable condition, which does not require ax-10 2136, ax-11 2151, ax-12 2167, ax-13 2383. Instance of nfbidv 1914. Note that the version of axc15 2438 with a disjoint variable condition is actually ax12v2 2169 (up to adding a superfluous antecedent). (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓)) | ||
Theorem | bj-equs45fv 34031* | Version of equs45f 2477 with a disjoint variable condition, which does not require ax-13 2383. Note that the version of equs5 2478 with a disjoint variable condition is actually sb56 2269 (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | bj-hbs1 34032* | Version of hbsb2 2517 with a disjoint variable condition, which does not require ax-13 2383, and removal of ax-13 2383 from hbs1 2266. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1v 34033* | Version of nfsb2 2518 with a disjoint variable condition, which does not require ax-13 2383, and removal of ax-13 2383 from nfs1v 2265. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | ||
Theorem | bj-hbsb2av 34034* | Version of hbsb2a 2519 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-hbsb3v 34035* | Version of hbsb3 2522 with a disjoint variable condition, which does not require ax-13 2383. (Remark: the unbundled version of nfs1 2523 is given by bj-nfs1v 34033.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfsab1 34036* | Remove dependency on ax-13 2383 from nfsab1 2808. UPDATE / TODO: nfsab1 2808 does not use ax-13 2383 either anymore; bj-nfsab1 34036 is shorter than nfsab1 2808 but uses ax-12 2167. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | ||
Theorem | bj-dtru 34037* |
Remove dependency on ax-13 2383 from dtru 5263. (Contributed by BJ,
31-May-2019.)
TODO: This predates the removal of ax-13 2383 in dtru 5263. But actually, sn-dtru 38991 is better than either, so move it to Main with sn-el 38990 (and determine whether bj-dtru 34037 should be kept as ALT or deleted). (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | bj-dtrucor2v 34038* | Version of dtrucor2 5265 with a disjoint variable condition, which does not require ax-13 2383 (nor ax-4 1801, ax-5 1902, ax-7 2006, ax-12 2167). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦) ⇒ ⊢ (𝜑 ∧ ¬ 𝜑) | ||
The closed formula ∀𝑥∀𝑦𝑥 = 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. | ||
Theorem | bj-hbaeb2 34039 | Biconditional version of a form of hbae 2448 with commuted quantifiers, not requiring ax-11 2151. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦) | ||
Theorem | bj-hbaeb 34040 | Biconditional version of hbae 2448. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦) | ||
Theorem | bj-hbnaeb 34041 | Biconditional version of hbnae 2449 (to replace it?). (Contributed by BJ, 6-Oct-2018.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | bj-dvv 34042 | A special instance of bj-hbaeb2 34039. 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 ↔ ∀𝑥∀𝑦 𝑥 = 𝑦) | ||
As a rule of thumb, if a theorem of the form ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (𝜒 ↔ 𝜃) is in the database, and the "more precise" theorems ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜒 → 𝜃) and ⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜃 → 𝜒) also hold (see bj-bisym 33822), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2434 (and equsalh 2436 and equsexh 2437). Even if only one of these two theorems holds, it should be added to the database. | ||
Theorem | bj-equsal1t 34043 | Duplication of wl-equsal1t 34663, with shorter proof. If one imposes a disjoint variable condition on x,y , then one can use alequexv 1998 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 34664 is also interesting. (Contributed by BJ, 6-Oct-2018.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) | ||
Theorem | bj-equsal1ti 34044 | Inference associated with bj-equsal1t 34043. (Contributed by BJ, 30-Sep-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑) | ||
Theorem | bj-equsal1 34045 | One direction of equsal 2433. (Contributed by BJ, 30-Sep-2018.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓) | ||
Theorem | bj-equsal2 34046 | One direction of equsal 2433. (Contributed by BJ, 30-Sep-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)) | ||
Theorem | bj-equsal 34047 | Shorter proof of equsal 2433. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2433, but "min */exc equsal" is ok. (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
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". | ||
Theorem | stdpc5t 34048 | Closed form of stdpc5 2199. (Possible to place it before 19.21t 2197 and use it to prove 19.21t 2197). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-stdpc5 34049 | More direct proof of stdpc5 2199. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) | ||
Theorem | 2stdpc5 34050 | A double stdpc5 2199 (one direction of PM*11.3). See also 2stdpc4 2066 and 19.21vv 40588. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓)) | ||
Theorem | bj-19.21t0 34051 | Proof of 19.21t 2197 from stdpc5t 34048. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | ||
Theorem | exlimii 34052 | Inference associated with exlimi 2208. Inferring a theorem when it is implied by an antecedent which may be true. (Contributed by BJ, 15-Sep-2018.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝜑 → 𝜓) & ⊢ ∃𝑥𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | ax11-pm 34053 | Proof of ax-11 2151 similar to PM's proof of alcom 2153 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 34057. Axiom ax-11 2151 is used in the proof only through nfa2 2166. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | ax6er 34054 | Commuted form of ax6e 2394. (Could be placed right after ax6e 2394). (Contributed by BJ, 15-Sep-2018.) |
⊢ ∃𝑥 𝑦 = 𝑥 | ||
Theorem | exlimiieq1 34055 | Inferring a theorem when it is implied by an equality which may be true. (Contributed by BJ, 30-Sep-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | exlimiieq2 34056 | 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.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | ax11-pm2 34057* | Proof of ax-11 2151 from the standard axioms of predicate calculus, similar to PM's proof of alcom 2153 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 2151 is used in the proof only through nfal 2334, nfsb 2561, sbal 2156, sb8 2555. See also ax11-pm 34053. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | bj-sbsb 34058 | Biconditional showing two possible (dual) definitions of substitution df-sb 2061 not using dummy variables. (Contributed by BJ, 19-Mar-2021.) |
⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∨ (𝑥 = 𝑦 ∧ 𝜑))) | ||
Theorem | bj-dfsb2 34059 | Alternate (dual) definition of substitution df-sb 2061 not using dummy variables. (Contributed by BJ, 19-Mar-2021.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∨ (𝑥 = 𝑦 ∧ 𝜑))) | ||
Theorem | bj-sbf3 34060 | Substitution has no effect on a bound variable (existential quantifier case); see sbf2 2263. (Contributed by BJ, 2-May-2019.) |
⊢ ([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑) | ||
Theorem | bj-sbf4 34061 | Substitution has no effect on a bound variable (nonfreeness case); see sbf2 2263. (Contributed by BJ, 2-May-2019.) |
⊢ ([𝑦 / 𝑥]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜑) | ||
Theorem | bj-sbnf 34062* | Move nonfree predicate in and out of substitution; see sbal 2156 and sbex 2280. (Contributed by BJ, 2-May-2019.) |
⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑) | ||
Theorem | bj-eu3f 34063* | Version of eu3v 2651 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 2651. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Miscellaneous theorems of first-order logic. | ||
Theorem | bj-sblem1 34064* | Lemma for substitution. (Contributed by BJ, 23-Jul-2023.) |
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜒))) | ||
Theorem | bj-sblem2 34065* | Lemma for substitution. (Contributed by BJ, 23-Jul-2023.) |
⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → ((∃𝑥𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-sblem 34066* | Lemma for substitution. (Contributed by BJ, 23-Jul-2023.) |
⊢ (∀𝑥(𝜑 → (𝜓 ↔ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜒))) | ||
Theorem | bj-sbievw1 34067* | Lemma for substitution. (Contributed by BJ, 23-Jul-2023.) |
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → 𝜓)) | ||
Theorem | bj-sbievw2 34068* | Lemma for substitution. (Contributed by BJ, 23-Jul-2023.) |
⊢ ([𝑦 / 𝑥](𝜓 → 𝜑) → (𝜓 → [𝑦 / 𝑥]𝜑)) | ||
Theorem | bj-sbievw 34069* | Lemma for substitution. Closed form of equsalvw 2001 and sbievw 2094. (Contributed by BJ, 23-Jul-2023.) |
⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) | ||
Theorem | bj-sbievv 34070 | Version of sbie 2540 with a second nonfreeness hypothesis and shorter proof. (Contributed by BJ, 18-Jul-2023.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | bj-moeub 34071 | Uniqueness is equivalent to existence being equivalent to unique existence. (Contributed by BJ, 14-Oct-2022.) |
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑)) | ||
Theorem | bj-sbidmOLD 34072 | Obsolete proof of sbidm 2548 temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | bj-dvelimdv 34073* |
Deduction form of dvelim 2468 with disjoint variable conditions. Uncurried
(imported) form of bj-dvelimdv1 34074. Typically, 𝑧 is a fresh
variable used for the implicit substitution hypothesis that results in
𝜒 (namely, 𝜓 can be thought as 𝜓(𝑥, 𝑦) and 𝜒 as
𝜓(𝑥, 𝑧)). So the theorem says that if x is
effectively free
in 𝜓(𝑥, 𝑧), then if x and y are not the same
variable, then
𝑥 is also effectively free in 𝜓(𝑥, 𝑦), in a context
𝜑.
One can weaken the implicit substitution hypothesis by adding the antecedent 𝜑 but this typically does not make the theorem much more useful. Similarly, one could use nonfreeness hypotheses instead of disjoint variable conditions but since this result is typically used when 𝑧 is a dummy variable, this would not be of much benefit. One could also remove DV (𝑥, 𝑧) since in the proof nfv 1906 can be replaced with nfal 2334 followed by nfn 1848. Remark: nfald 2339 uses ax-11 2151; it might be possible to inline and use ax11w 2125 instead, but there is still a use via 19.12 2338 anyway. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓)) ⇒ ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | ||
Theorem | bj-dvelimdv1 34074* | Curried (exported) form of bj-dvelimdv 34073 (of course, one is directly provable from the other, but we keep this proof for illustration purposes). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)) | ||
Theorem | bj-dvelimv 34075* | A version of dvelim 2468 using the "nonfree" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜑)) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑) | ||
Theorem | bj-nfeel2 34076* | Nonfreeness in a membership statement. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧) | ||
Theorem | bj-axc14nf 34077 | Proof of a version of axc14 2481 using the "nonfree" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦)) | ||
Theorem | bj-axc14 34078 | Alternate proof of axc14 2481 (even when inlining the above results, this gives a shorter proof). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))) | ||
Theorem | mobidvALT 34079* | Alternate proof of mobidv 2629 directly from its analogues albidv 1912 and exbidv 1913, using deduction style. Note the proof structure, similar to mobi 2626. (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. Remove dependency on ax-6 1961, ax-7 2006, ax-12 2167 by adapting proof of mobid 2630. (Revised by BJ, 26-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
In this section, we give a sketch of the proof of the Eliminability Theorem for class terms in an extensional set theory where quantification occurs only over set variables. Eliminability of class variables using the $a-statements ax-ext 2793, df-clab 2800, df-cleq 2814, df-clel 2893 is an easy result, proved for instance in Appendix X of Azriel Levy, Basic Set Theory, Dover Publications, 2002. Note that viewed from the set.mm axiomatization, it is a metatheorem not formalizable is set.mm. It states: every formula in the language of FOL + ∈ + class terms, but without class variables, is provably equivalent (over {FOL, ax-ext 2793, df-clab 2800, df-cleq 2814, df-clel 2893 }) to a formula in the language of FOL + ∈ (that is, without class terms). The proof goes by induction on the complexity of the formula (see op. cit. for details). The base case is that of atomic formulas. The atomic formulas containing class terms are of one of the following forms: for equality, 𝑥 = {𝑦 ∣ 𝜑}, {𝑥 ∣ 𝜑} = 𝑦, {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}, and for membership, 𝑦 ∈ {𝑥 ∣ 𝜑}, {𝑥 ∣ 𝜑} ∈ 𝑦, {𝑥 ∣ 𝜑} ∈ {𝑦 ∣ 𝜓}. These cases are dealt with by eliminable1 34080 and the following theorems of this section, which are special instances of df-clab 2800, dfcleq 2815 (proved from {FOL, ax-ext 2793, df-cleq 2814 }), and df-clel 2893. Indeed, denote by (i) the formula proved by "eliminablei". One sees that the RHS of (1) has no class terms, the RHS's of (2x) have only class terms of the form dealt with by (1), and the RHS's of (3x) have only class terms of the forms dealt with by (1) and (2a). Note that in order to prove eliminable2a 34081, eliminable2b 34082 and eliminable3a 34084, we need to substitute a class variable for a setvar variable. This is possible because setvars are class terms: this is the content of the syntactic theorem cv 1527, which is used in these proofs (this does not appear in the html pages but it is in the set.mm file and you can check it using the Metamath program). The induction step relies on the fact that any formula is a FOL-combination of atomic formulas, so if one found equivalents for all atomic formulas constituting the formula, then the same FOL-combination of these equivalents will be equivalent to the original formula. Note that one has a slightly more precise result: if the original formula has only class terms appearing in atomic formulas of the form 𝑦 ∈ {𝑥 ∣ 𝜑}, then df-clab 2800 is sufficient (over FOL) to eliminate class terms, and if the original formula has only class terms appearing in atomic formulas of the form 𝑦 ∈ {𝑥 ∣ 𝜑} and equalities, then df-clab 2800, ax-ext 2793 and df-cleq 2814 are sufficient (over FOL) to eliminate class terms. To prove that { df-clab 2800, df-cleq 2814, df-clel 2893 } provides a definitional extension of {FOL, ax-ext 2793 }, one needs to prove the above Eliminability Theorem, which compares the expressive powers of the languages with and without class terms, and the Conservativity Theorem, which compares the deductive powers when one adds { df-clab 2800, df-cleq 2814, df-clel 2893 }. It states that a formula without class terms is provable in one axiom system if and only if it is provable in the other, and that this remains true when one adds further definitions to {FOL, ax-ext 2793 }. It is also proved in op. cit. The proof is more difficult, since one has to construct for each proof of a statement without class terms, an associated proof not using { df-clab 2800, df-cleq 2814, df-clel 2893 }. It involves a careful case study on the structure of the proof tree. | ||
Theorem | eliminable1 34080 | A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | eliminable2a 34081* | A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = {𝑦 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑})) | ||
Theorem | eliminable2b 34082* | A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ({𝑥 ∣ 𝜑} = 𝑦 ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ 𝑦)) | ||
Theorem | eliminable2c 34083* | A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ({𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓})) | ||
Theorem | eliminable3a 34084* | A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ({𝑥 ∣ 𝜑} ∈ 𝑦 ↔ ∃𝑧(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 ∈ 𝑦)) | ||
Theorem | eliminable3b 34085* | A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ({𝑥 ∣ 𝜑} ∈ {𝑦 ∣ 𝜓} ↔ ∃𝑧(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 ∈ {𝑦 ∣ 𝜓})) | ||
A few results about classes can be proved without using ax-ext 2793. One could move all theorems from cab 2799 to df-clel 2893 (except for dfcleq 2815 and cvjust 2816) in a subsection "Classes" before the subsection on the axiom of extensionality, together with the theorems below. In that subsection, the last statement should be df-cleq 2814. Note that without ax-ext 2793, the $a-statements df-clab 2800, df-cleq 2814, and df-clel 2893 are no longer eliminable (see previous section) (but PROBABLY df-clab 2800 is still conservative , while df-cleq 2814 and df-clel 2893 are not). This is not a reason not to study what is provable with them but without ax-ext 2793, in order to gauge their strengths more precisely. Before that subsection, a subsection "The membership predicate" could group the statements with ∈ that are currently in the FOL part (including wcel 2105, wel 2106, ax-8 2107, ax-9 2115). Remark: the weakening of eleq1 2900 / eleq2 2901 to eleq1w 2895 / eleq2w 2896 can also be done with eleq1i 2903, eqeltri 2909, eqeltrri 2910, eleq1a 2908, eleq1d 2897, eqeltrd 2913, eqeltrrd 2914, eqneltrd 2932, eqneltrrd 2933, nelneq 2937. Remark: possibility to remove dependency on ax-10 2136, ax-11 2151, ax-13 2383 from nfcri 2971 and theorems using it if one adds a disjoint variable condition (that theorem is typically used with dummy variables, so the disjoint variable condition addition is not very restrictive), and then shorten nfnfc 2990. | ||
Theorem | bj-denotes 34086* |
This would be the justification theorem for the definition of the unary
predicate "E!" by ⊢ ( E! 𝐴 ↔ ∃𝑥𝑥 = 𝐴) which could be
interpreted as "𝐴 exists" (as a set) or
"𝐴 denotes" (in the
sense of free logic).
A shorter proof using bitri 276 (to add an intermediate proposition ∃𝑧𝑧 = 𝐴 with a fresh 𝑧), cbvexvw 2035, and eqeq1 2825, requires the core axioms and { ax-9 2115, ax-ext 2793, df-cleq 2814 } whereas this proof requires the core axioms and { ax-8 2107, df-clab 2800, df-clel 2893 }. Theorem bj-issetwt 34087 proves that "existing" is equivalent to being a member of a class abstraction. It also requires, with the present proof, { ax-8 2107, df-clab 2800, df-clel 2893 } (whereas with the shorter proof from cbvexvw 2035 and eqeq1 2825 it would require { ax-8 2107, ax-9 2115, ax-ext 2793, df-clab 2800, df-cleq 2814, df-clel 2893 }). That every class is equal to a class abstraction is proved by abid1 2956, which requires { ax-8 2107, ax-9 2115, ax-ext 2793, df-clab 2800, df-cleq 2814, df-clel 2893 }. Note that there is no disjoint variable condition on 𝑥, 𝑦 but the theorem does not depend on ax-13 2383. Actually, the proof depends only on the logical axioms ax-1 6 through ax-7 2006 and sp 2172. The symbol "E!" was chosen to be reminiscent of the analogous predicate in (inclusive or non-inclusive) free logic, which deals with the possibility of nonexistent objects. This analogy should not be taken too far, since here there are no equality axioms for classes: these are derived from ax-ext 2793 and df-cleq 2814 (e.g., eqid 2821 and eqeq1 2825). In particular, one cannot even prove ⊢ ∃𝑥𝑥 = 𝐴 ⇒ ⊢ 𝐴 = 𝐴 without ax-ext 2793 and df-cleq 2814. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) |
⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) | ||
Theorem | bj-issetwt 34087* | Closed form of bj-issetw 34088. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)) | ||
Theorem | bj-issetw 34088* | The closest one can get to isset 3507 without using ax-ext 2793. See also vexw 2805. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3507 using eleq2i 2904 (which requires ax-ext 2793 and df-cleq 2814). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) |
⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴) | ||
Theorem | bj-elissetv 34089* | Version of bj-elisset 34090 with a disjoint variable condition on 𝑥, 𝑉. This proof uses only df-ex 1772, ax-gen 1787, ax-4 1801 and df-clel 2893 on top of propositional calculus. Prefer its use over bj-elisset 34090 when sufficient. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | ||
Theorem | bj-elisset 34090* | Remove from elisset 3506 dependency on ax-ext 2793 (and on df-cleq 2814 and df-v 3497). This proof uses only df-clab 2800 and df-clel 2893 on top of first-order logic. It only requires ax-1--7 and sp 2172. Use bj-elissetv 34089 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | ||
Theorem | bj-issetiv 34091* | Version of bj-isseti 34092 with a disjoint variable condition on 𝑥, 𝑉. This proof uses only df-ex 1772, ax-gen 1787, ax-4 1801 and df-clel 2893 on top of propositional calculus. Prefer its use over bj-isseti 34092 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝑉 ⇒ ⊢ ∃𝑥 𝑥 = 𝐴 | ||
Theorem | bj-isseti 34092* | Remove from isseti 3509 dependency on ax-ext 2793 (and on df-cleq 2814 and df-v 3497). This proof uses only df-clab 2800 and df-clel 2893 on top of first-order logic. It only uses ax-12 2167 among the auxiliary logical axioms. The hypothesis uses 𝑉 instead of V for extra generality. This is indeed more general as long as elex 3513 is not available. Use bj-issetiv 34091 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝑉 ⇒ ⊢ ∃𝑥 𝑥 = 𝐴 | ||
Theorem | bj-ralvw 34093 | A weak version of ralv 3520 not using ax-ext 2793 (nor df-cleq 2814, df-clel 2893, df-v 3497), and only core FOL axioms. See also bj-rexvw 34094. The analogues for reuv 3522 and rmov 3523 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ 𝜓 ⇒ ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | bj-rexvw 34094 | A weak version of rexv 3521 not using ax-ext 2793 (nor df-cleq 2814, df-clel 2893, df-v 3497), and only core FOL axioms. See also bj-ralvw 34093. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ 𝜓 ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑) | ||
Theorem | bj-rababw 34095 | A weak version of rabab 3524 not using df-clel 2893 nor df-v 3497 (but requiring ax-ext 2793) nor ax-12 2167. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ 𝜓 ⇒ ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑} | ||
Theorem | bj-rexcom4bv 34096* | Version of rexcom4b 3525 and bj-rexcom4b 34097 with a disjoint variable condition on 𝑥, 𝑉, hence removing dependency on df-sb 2061 and df-clab 2800 (so that it depends on df-clel 2893 and df-rex 3144 only on top of first-order logic). Prefer its use over bj-rexcom4b 34097 when sufficient (in particular when 𝑉 is substituted for V). Note the 𝑉 in the hypothesis instead of V. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.) |
⊢ 𝐵 ∈ 𝑉 ⇒ ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) | ||
Theorem | bj-rexcom4b 34097* | Remove from rexcom4b 3525 dependency on ax-ext 2793 and ax-13 2383 (and on df-or 842, df-cleq 2814, df-nfc 2963, df-v 3497). The hypothesis uses 𝑉 instead of V (see bj-isseti 34092 for the motivation). Use bj-rexcom4bv 34096 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ 𝐵 ∈ 𝑉 ⇒ ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) | ||
Theorem | bj-ceqsalt0 34098 | The FOL content of ceqsalt 3528. Lemma for bj-ceqsalt 34100 and bj-ceqsaltv 34101. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜃 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥𝜃) → (∀𝑥(𝜃 → 𝜑) ↔ 𝜓)) | ||
Theorem | bj-ceqsalt1 34099 | The FOL content of ceqsalt 3528. Lemma for bj-ceqsalt 34100 and bj-ceqsaltv 34101. TODO: consider removing if it does not add anything to bj-ceqsalt0 34098. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.) |
⊢ (𝜃 → ∃𝑥𝜒) ⇒ ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓)) | ||
Theorem | bj-ceqsalt 34100* | Remove from ceqsalt 3528 dependency on ax-ext 2793 (and on df-cleq 2814 and df-v 3497). Note: this is not doable with ceqsralt 3529 (or ceqsralv 3534), which uses eleq1 2900, but the same dependence removal is possible for ceqsalg 3530, ceqsal 3532, ceqsalv 3533, cgsexg 3538, cgsex2g 3539, cgsex4g 3540, ceqsex 3541, ceqsexv 3542, ceqsex2 3544, ceqsex2v 3545, ceqsex3v 3546, ceqsex4v 3547, ceqsex6v 3548, ceqsex8v 3549, gencbvex 3550 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3551, gencbval 3552, vtoclgft 3554 (it uses Ⅎ, whose justification nfcjust 2962 does not use ax-ext 2793) and several other vtocl* theorems (see for instance bj-vtoclg1f 34132). See also bj-ceqsaltv 34101. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
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