P. 372 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37101-37200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-ssbid1ALT 37101	Alternate proof of bj-ssbid1 37100, not using sbequ1 2282. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → [𝑥 / 𝑥]𝜑)
Theorem	bj-ax6elem1 37102*	Lemma for bj-ax6e 37104. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	bj-ax6elem2 37103*	Lemma for bj-ax6e 37104. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Theorem	bj-ax6e 37104	Proof of ax6e 2413 (hence ax6 2414) from Tarski's system, ax-c9 39478, ax-c16 39480. Remark: ax-6 1986 is used only via its principal (unbundled) instance ax6v 1987. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦
21.19.4.6  Adding ax-6
Theorem	bj-spim0 37105*	A universal specialization result in deduction form, proved from ax-1 6 -- ax-6 1986, where the only DV condition is on 𝑥, 𝑦 and where 𝑥 should be nonfree in the new proposition 𝜒 (and in the context 𝜑). (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-spimvwt 37106*	Closed form of spimvw 2005. See also spimt 2416. (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-spnfw 37107	Theorem close to a closed form of spnfw 1998. (Contributed by BJ, 12-May-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbvexiw 37108*	Change bound variable. This is to cbvexvw 2056 what cbvaliw 2025 is to cbvalvw 2055. TODO: move after cbvalivw 2026. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-cbvexivw 37109*	Change bound variable. This is to cbvexvw 2056 what cbvalivw 2026 is to cbvalvw 2055. TODO: move after cbvalivw 2026. (Contributed by BJ, 17-Mar-2020.)
⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
Theorem	bj-modald 37110	A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	bj-denot 37111*	A weakening of ax-6 1986 and ax6v 1987. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.)
⊢ (𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥)
Theorem	bj-eqs 37112*	A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2402. The disjoint variable condition DV (𝑥, 𝜑) is necessary for both directions: consider substituting 𝑥 = 𝑧 for 𝜑. (Contributed by BJ, 25-May-2021.)
⊢ (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
21.19.4.7  Adding ax-7
Theorem	bj-cbvexw 37113*	Change bound variable. This is to cbvexvw 2056 what cbvalw 2054 is to cbvalvw 2055. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Theorem	bj-ax12w 37114*	The general statement that ax12w 2166 proves. (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓)))
21.19.4.8  Membership predicate, ax-8 and ax-9
Theorem	bj-ax89 37115	A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2143 and ax-9 2151. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2143 and ax-9 2151, as proved here. In the other direction, one can prove ax-8 2143 (respectively ax-9 2151) from bj-ax89 37115 by using mpan2 701 (respectively mpan 700) and equid 2031. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))
Theorem	bj-cleljusti 37116*	One direction of cleljust 2150, requiring only ax-1 6-- ax-5 1929 and ax8v1 2145. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
21.19.4.9  Adding ax-11
Theorem	bj-alcomexcom 37117	Commutation of two existential quantifiers on a formula is equivalent to commutation of two universal quantifiers over the same variables on the negation of that formula. Can be placed in the ax-4 1828 section, soon after 2nexaln 1849, and used to prove excom 2195. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))
Theorem	bj-hbald 37118	General statement that hbald 2201 proves . (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑦𝜓)    &   ⊢ (𝜓 → (𝜒 → ∀𝑥𝜃))    ⇒   ⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑥∀𝑦𝜃))
Theorem	bj-hbalt 37119	Closed form of (general instance of) hbal 2200. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑦(𝜑 → ∀𝑥𝜓) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-hbal 37120	More general instance of hbal 2200. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)
21.19.4.10  Adding ax-12
Theorem	axc11n11 37121	Proof of axc11n 2456 from { ax-1 6-- ax-7 2027, axc11 2460 } . Almost identical to axc11nfromc11 39514. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	axc11n11r 37122	Proof of axc11n 2456 from { ax-1 6-- ax-7 2027, axc9 2412, axc11r 2398 } (note that axc16 2295 is provable from { ax-1 6-- ax-7 2027, axc11r 2398 }). Note that axc11n 2456 proves (over minimal calculus) that axc11 2460 and axc11r 2398 are equivalent. Therefore, axc11n11 37121 and axc11n11r 37122 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2460 appears slightly stronger since axc11n11r 37122 requires axc9 2412 while axc11n11 37121 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	bj-axc16g16 37123*	Proof of axc16g 2294 from { ax-1 6-- ax-7 2027, axc16 2295 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	bj-ax12v3 37124*	A weak version of ax-12 2211 which is stronger than ax12v 2212. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2031), then bj-ax12v3 37124 implies ax-5 1929 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 37125. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-ax12v3ALT 37125*	Alternate proof of bj-ax12v3 37124. Uses axc11r 2398 and axc15 2452 instead of ax-12 2211. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-sb 37126*	A weak variant of sbid2 2538 not requiring ax-13 2402 nor ax-10 2174. On top of Tarski's FOL, one implication requires only ax12v 2212, and the other requires only sp 2217. (Contributed by BJ, 25-May-2021.)
⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-modalbe 37127	The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2350. (Contributed by BJ, 20-Oct-2019.)
⊢ (𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	bj-spst 37128	Closed form of sps 2219. Once in main part, prove sps 2219 and spsd 2221 from it. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-19.21bit 37129	Closed form of 19.21bi 2223. (Contributed by BJ, 20-Oct-2019.)
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
Theorem	bj-19.23bit 37130	Closed form of 19.23bi 2225. (Contributed by BJ, 20-Oct-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓))
Theorem	bj-nexrt 37131	Closed form of nexr 2226. Contrapositive of 19.8a 2215. (Contributed by BJ, 20-Oct-2019.)
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
Theorem	bj-alrim 37132	Closed form of alrimi 2247. (Contributed by BJ, 2-May-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-alrim2 37133	Uncurried (imported) form of bj-alrim 37132. (Contributed by BJ, 2-May-2019.)
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓))
Theorem	bj-nfdt0 37134	A theorem close to a closed form of nf5d 2317 and nf5dh 2180. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))
Theorem	bj-nfdt 37135	Closed form of nf5d 2317 and nf5dh 2180. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓)))
Theorem	bj-nexdt 37136	Closed form of nexd 2255. (Contributed by BJ, 20-Oct-2019.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdvt 37137*	Closed form of nexdv 1955. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))
Theorem	bj-alexbiex 37138	Adding a second quantifier over the same variable is a transparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-exexbiex 37139	Adding a second quantifier over the same variable is a transparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∃𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-alalbial 37140	Adding a second quantifier over the same variable is a transparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	bj-exalbial 37141	Adding a second quantifier over the same variable is a transparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.)
⊢ (∃𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	bj-19.9htbi 37142	Strengthening 19.9ht 2351 by replacing its consequent with a biconditional (19.9t 2238 does have a biconditional consequent). This propagates. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))
Theorem	bj-hbntbi 37143	Strengthening hbnt 2327 by replacing its consequent with a biconditional. See also hbntg 36117 and hbntal 45093. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 37142. (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
Theorem	bj-biexal1 37144	A general FOL biconditional that generalizes 19.9ht 2351 among others. For this and the following theorems, see also 19.35 1896, 19.21 2241, 19.23 2245. When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexal2 37145	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(∃𝑥𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexal3 37146	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑 → 𝜓))
Theorem	bj-bialal 37147	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-biexex 37148	When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
⊢ (∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-hbexd 37149	A more general instance of the deduction form of hbex 2356. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑦𝜓)    &   ⊢ (𝜓 → (𝜒 → ∀𝑥𝜃))    ⇒   ⊢ (𝜑 → (∃𝑦𝜒 → ∀𝑥∃𝑦𝜃))
Theorem	bj-hbext 37150	Closed form of bj-hbex 37151 and hbex 2356. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓))
Theorem	bj-hbex 37151	A more general instance of hbex 2356. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓)
Theorem	bj-nfalt 37152	Closed form of nfal 2354. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)
Theorem	bj-nfext 37153	Closed form of nfex 2355. (Contributed by BJ, 10-Oct-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)
Theorem	bj-eeanvw 37154*	Version of exdistrv 1974 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2190. (The same can be done with eeeanv 2380 and ee4anv 2381.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	bj-modal4 37155	First-order logic form of the modal axiom (4). See hba1 2326. This is the standard proof of the implication in modal logic (B5 ⇒ 4). Its dual statement is bj-modal4e 37156. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	bj-modal4e 37156	First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 37155 (hba1 2326). (Contributed by BJ, 21-Dec-2020.) (Proof modification is discouraged.)
⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)
Theorem	bj-modalb 37157	A short form of the axiom B of modal logic using only primitive symbols (→ , ¬ , ∀). (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	bj-wnf1 37158	When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-wnf2 37159	When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-wnfanf 37160	When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the universal form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
Theorem	bj-wnfenf 37161	When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the existential form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → 𝜓))
Theorem	bj-19.12 37162	See 19.12 2358. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2195 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1803 or df-bj-nnf 37166, directly or indirectly. (Proof modification is discouraged.)
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)
21.19.4.11  Really adding ax-12 The results in the previous section, as actually many theorems of the main part using ax-12 2211, actually only require sp 2217 (which is proved using ax-12 2211).
Theorem	bj-substax12 37163	Equivalent form of the axiom of substitution bj-ax12 37093. Although both sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 37124 on 𝑡, 𝜑) to hold, their equivalence holds without DV conditions. The forward implication is proved in modal (K4) while the reverse implication is proved in modal (T5). The LHS has the advantage of not involving nested quantifiers on the same variable. Its metaweakening is proved from the core axiom schemes in bj-substw 37164. Note that in the LHS, the reverse implication holds by equs4 2446 (or equs4v 2019 if a DV condition is added on 𝑥, 𝑡 as in bj-ax12 37093), and the forward implication is sbalex 2276. The LHS can be read as saying that if there exists a variable equal to a given term witnessing a given formula, then all variables equal to that term also witness that formula. The equivalent form of the LHS using only primitive symbols is (∀𝑥(𝑥 = 𝑡 → 𝜑) ∨ ∀𝑥(𝑥 = 𝑡 → ¬ 𝜑)), which expresses that a given formula is true at all variables equal to a given term, or false at all these variables. An equivalent form of the LHS using only the existential quantifier is ¬ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑡 ∧ ¬ 𝜑)), which expresses that there can be no two variables both equal to a given term, one witnessing a formula and the other witnessing its negation. These equivalences do not hold in intuitionistic logic. The LHS should be the preferred form, and has the advantage of having no negation nor nested quantifiers. (Contributed by BJ, 21-May-2024.) (Proof modification is discouraged.)
⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
Theorem	bj-substw 37164*	Weak form of the LHS of bj-substax12 37163 proved from the core axiom schemes. Compare ax12w 2166. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))
21.19.4.12  Nonfreeness
Syntax	wnnf 37165	Syntax for the nonfreeness quantifier.
wff Ⅎ'𝑥𝜑
Definition	df-bj-nnf 37166	Definition of the nonfreeness quantifier. The formula Ⅎ'𝑥𝜑 has the intended meaning that the variable 𝑥 is semantically nonfree in the formula 𝜑. The motivation for this quantifier is to have a condition expressible in the logic which is as close as possible to the non-occurrence condition DV (𝑥, 𝜑) (in Metamath files, "$d x ph $."), which belongs to the metalogic. The standard syntactic nonfreeness condition, also expressed in the metalogic, is intermediate between these two notions: semantic nonfreeness implies syntactic nonfreeness, which implies non-occurrence. Both implications are strict; for the first, note that ⊢ Ⅎ'𝑥𝑥 = 𝑥, that is, 𝑥 is semantically (but not syntactically) nonfree in the formula 𝑥 = 𝑥; for the second, note that 𝑥 is syntactically nonfree in the formula ∀𝑥𝑥 = 𝑥 although it occurs in it. We now prove two metatheorems which make precise the above fact that, as far as proving power is concerned, the nonfreeness condition Ⅎ'𝑥𝜑 is very close to the non-occurrence condition DV (𝑥, 𝜑). Let S be a Metamath system with the FOL-syntax of (i)set.mm, containing intuitionistic positive propositional calculus and ax-5 1929 and ax5e 1931. Theorem 1. If the scheme (Ⅎ'𝑥𝜑 & PHI1 & ... & PHIn ⇒ PHI0, DV) is provable in S, then so is the scheme (PHI1 & ... & PHIn ⇒ PHI0, DV ∪ {{𝑥, 𝜑}}). Proof: By bj-nnfv 37207, we can prove (Ⅎ'𝑥𝜑, {{𝑥, 𝜑}}), from which the theorem follows. QED Theorem 2. Suppose that S also contains (the FOL version of) modal logic KB and commutation of quantifiers alcom 2192 and excom 2195 (possibly weakened by a DV condition on the quantifying variables), and that S can be axiomatized such that the only axioms with a DV condition involving a formula variable are among ax-5 1929, ax5e 1931, ax5ea 1932. If the scheme (PHI1 & ... & PHIn ⇒ PHI0, DV) is provable in S, then so is the scheme (Ⅎ'𝑥𝜑 & PHI1 & ... & PHIn ⇒ PHI0, DV ∖ {{𝑥, 𝜑}}). More precisely, if S contains modal 45 and if the variables quantified over in PHI0, ..., PHIn are among 𝑥1, ..., 𝑥m, then the scheme (PHI1 & ... & PHIn ⇒ (antecedent → PHI0), DV ∖ {{𝑥, 𝜑}}) is provable in S, where the antecedent is a finite conjunction of formulas of the form ∀𝑥i1 ...∀𝑥ip Ⅎ'𝑥𝜑 where the 𝑥ij's are among the 𝑥i's. Lemma: If 𝑥 ∉ OC(PHI), then S proves the scheme (Ⅎ'𝑥𝜑 ⇒ Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}). More precisely, if the variables quantified over in PHI are among 𝑥1, ..., 𝑥m, then ((antecedent → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) is provable in S, with the same form of antecedent as above. Proof: By induction on the height of PHI. We first note that by bj-nnfbi 37186 we can assume that PHI contains only primitive (as opposed to defined) symbols. For the base case, atomic formulas are either 𝜑, in which case the scheme to prove is an instance of id 22, or have variables all in OC(PHI) ∖ {𝜑}, so (Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) by bj-nnfv 37207, hence ((Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) by a1i 11. For the induction step, PHI is either an implication, a negation, a conjunction, a disjunction, a biconditional, a universal or an existential quantification of formulas where 𝑥 does not occur. We use respectively bj-nnfim 37191, bj-nnfnt 37189, bj-nnfan 37193, bj-nnfor 37195, bj-nnfbit 37197, bj-nnfalt 37229, bj-nnfext 37230. For instance, in the implication case, if we have by induction hypothesis ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}) and ((∀𝑦1 ...∀𝑦n Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PSI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) ∖ {𝜑}}), then bj-nnfim 37191 yields (((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 ∧ ∀𝑦1 ...∀𝑦n Ⅎ'𝑥𝜑) → Ⅎ'𝑥 (PHI → PSI)), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI → PSI) ∖ {𝜑}}) and similarly for antecedents which are conjunctions as in the statement of the lemma. In the universal quantification case, say quantification over 𝑦, if we have by induction hypothesis ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PHI) ∖ {𝜑}}), then bj-nnfalt 37229 yields ((∀𝑦∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥∀𝑦 PHI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(∀𝑦 PHI) ∖ {𝜑}}) and similarly for antecedents which are conjunctions as in the statement of the lemma. Note bj-nnfalt 37229 and bj-nnfext 37230 are proved from positive propositional calculus with alcom 2192 and excom 2195 (possibly weakened by a DV condition on the quantifying variables), and modalB (via bj-19.12 37162). QED Proof of the theorem: Consider a proof of that scheme directly from the axioms. Consider a step where a DV condition involving 𝜑 is used. By hypothesis, that step is an instance of ax-5 1929 or ax5e 1931 or ax5ea 1932. It has the form (PSI → ∀𝑥 PSI) where PSI has the form of the lemma and the DV conditions of the proof contain {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) }. Therefore, one has ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PSI), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) ∖ {𝜑}}) for appropriate 𝑥i's, and by bj-nnfa 37167 we obtain ((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → (PSI → ∀𝑥 PSI)), {{𝑥, 𝑎} ∣ 𝑎 ∈ OC(PSI) ∖ {𝜑}}) and similarly for antecedents which are conjunctions as in the statement of the theorem. Similarly if the step is using ax5e 1931 or ax5ea 1932, we would use bj-nnfe 37170 or bj-nnfea 37173 respectively. Therefore, taking as antecedent of the theorem to prove the conjunction of all the antecedents at each of these steps, we obtain a proof by "carrying the context over", which is possible, as in the deduction theorem when the step uses ax-mp 5, and when the step uses ax-gen 1814, by bj-nnf-alrim 37184 and bj-nnfa1 37223 (which requires modal 45). The condition DV (𝑥, 𝜑) is not required by the resulting proof. Finally, there may be in the global antecedent thus constructed some dummy variables, which can be removed by spvw 2000. QED Compared with df-nf 1803, the present definition is stricter on positive propositional calculus (bj-nnfnfTEMP 37179) and equivalent on core FOL plus sp 2217 (bj-nfnnfTEMP 37221). While being stricter, it still holds for non-occurring variables (bj-nnfv 37207), which is the basic requirement for this quantifier. In particular, it translates more closely the associated variable disjointness condition. Since the nonfreeness quantifier is a means to translate a variable disjointness condition from the metalogic to the logic, it seems preferable. Also, since nonfreeness is mainly used as a hypothesis, this definition would allow more theorems, notably the 19.xx theorems, to be proved from the core axioms, without needing a 19.xxv variant. One can devise infinitely many definitions increasingly close to the non-occurring condition, like ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ ∀𝑥((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ ∀𝑥∀𝑥... and each stronger definition would permit more theorems to be proved from the core axioms. A reasonable rule seems to be to stop before nested quantifiers appear (since they typically require ax-10 2174 to work with), and also not to have redundant conjuncts when full metacomplete FOL= is developed. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
Theorem	bj-nnfa 37167	Nonfreeness implies the equivalent of ax-5 1929. See nf5r 2228. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Theorem	bj-nnfad 37168	Nonfreeness implies the equivalent of ax-5 1929, deduction form. See nf5rd 2230. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	bj-nnfai 37169	Nonfreeness implies the equivalent of ax-5 1929, inference form. See nf5ri 2229. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	bj-nnfe 37170	Nonfreeness implies the equivalent of ax5e 1931. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
Theorem	bj-nnfed 37171	Nonfreeness implies the equivalent of ax5e 1931, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓))
Theorem	bj-nnfei 37172	Nonfreeness implies the equivalent of ax5e 1931, inference form. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → 𝜑)
Theorem	bj-nnfea 37173	Nonfreeness implies the equivalent of ax5ea 1932. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nnfead 37174	Nonfreeness implies the equivalent of ax5ea 1932, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
Theorem	bj-nnfeai 37175	Nonfreeness implies the equivalent of ax5ea 1932, inference form. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
Theorem	bj-alnnf 37176	In deduction-style proofs, it is equivalent to assert that the context holds for all values of a variable, or that is does not depend on that variable. (Contributed by BJ, 28-Mar-2026.)
⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → Ⅎ'𝑥𝜑))
Theorem	bj-alnnf2 37177	If a proposition holds, then it holds for all values of a given variable if and only if it does not depend on that variable. (Contributed by BJ, 28-Mar-2026.)
⊢ (𝜑 → (∀𝑥𝜑 ↔ Ⅎ'𝑥𝜑))
Theorem	bj-dfnnf2 37178	Alternate definition of df-bj-nnf 37166 using only primitive symbols (→, ¬, ∀) in each conjunct. (Contributed by BJ, 20-Aug-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ ((𝜑 → ∀𝑥𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
Theorem	bj-nnfnfTEMP 37179	New nonfreeness implies old nonfreeness on minimal implicational calculus (the proof indicates it uses ax-3 8 because of set.mm's definition of the biconditional, but the proof actually holds in minimal implicational calculus). (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1803 except via df-nf 1803 directly. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	bj-nnfim1 37180	A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
Theorem	bj-nnfim2 37181	A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → 𝜓)))
Theorem	bj-nnftht 37182	A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2217 (modal T), as in bj-nnfbi 37186. (Contributed by BJ, 28-Jul-2023.)
⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)
Theorem	bj-nnfth 37183	A variable is nonfree in a theorem, inference form. (Contributed by BJ, 28-Jul-2023.)
⊢ 𝜑    ⇒   ⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnf-alrim 37184	Proof of the closed form of alrimi 2247 from modalK (compare alrimiv 1946). See also bj-alrim 37132. Actually, most proofs between 19.3t 2235 and 2sbbid 2281 could be proved without ax-12 2211. (Contributed by BJ, 20-Aug-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-stdpc5t 37185	Alias of bj-nnf-alrim 37184 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2242 proved from modalK (obsoleting stdpc5v 1957). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 37184 instead. (New usaged is discouraged.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-nnfbi 37186	If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other. Compare nfbiit 1870. From this and bj-nnfim 37191 and bj-nnfnt 37189, one can prove analogous nonfreeness conservation results for other propositional operators. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 37182) in order not to require sp 2217 (modal T). (Contributed by BJ, 27-Aug-2023.)
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))
Theorem	bj-nnfbd0 37187	If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other, deduction form. The antecedent of the conclusion is in the "strong necessity" modality of modal logic (see also bj-nnftht 37182) in order not to require sp 2217 (modal T). See bj-nnfbi 37186. (Contributed by BJ, 21-Mar-2026.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ ∀𝑥𝜑) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
Theorem	bj-nnfbii 37188	If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi 37186. (Contributed by BJ, 18-Nov-2023.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)
Theorem	bj-nnfnt 37189	A variable is nonfree in a formula if and only if it is nonfree in its negation. The foward implication is intuitionistically valid (and that direction is sufficient for the purpose of recursively proving that some formulas have a given variable not free in them, like bj-nnfim 37191). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1875. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)
Theorem	bj-nnfnth 37190	A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.)
⊢ ¬ 𝜑    ⇒   ⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnfim 37191	Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication. (Contributed by BJ, 27-Aug-2023.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓))
Theorem	bj-nnfimd 37192	Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication, deduction form. (Contributed by BJ, 2-Dec-2023.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 → 𝜒))
Theorem	bj-nnfan 37193	Nonfreeness in both conjuncts implies nonfreeness in the conjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of conjunction in terms of implication and negation, so using bj-nnfim 37191, bj-nnfnt 37189 and bj-nnfbi 37186, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∧ 𝜓))
Theorem	bj-nnfand 37194	Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 37193, it has two more essential steps but fewer total steps (since there are fewer intermediate formulas to build) and is easier to follow and understand. This statement is of intermediate complexity: for simpler statements, closed-style proofs like that of bj-nnfan 37193 will generally be shorter than deduction-style proofs while still easy to follow, while for more complex statements, the opposite will be true (and deduction-style proofs like that of bj-nnfand 37194 will generally be easier to understand). (Contributed by BJ, 19-Nov-2023.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∧ 𝜒))
Theorem	bj-nnfor 37195	Nonfreeness in both disjuncts implies nonfreeness in the disjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of disjunction in terms of implication and negation, so using bj-nnfim 37191, bj-nnfnt 37189 and bj-nnfbi 37186, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∨ 𝜓))
Theorem	bj-nnford 37196	Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 37195 and bj-nnfand 37194. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒))
Theorem	bj-nnfbit 37197	Nonfreeness in both sides implies nonfreeness in the biconditional. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ↔ 𝜓))
Theorem	bj-nnfbid 37198	Nonfreeness in both sides implies nonfreeness in the biconditional, deduction form. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ↔ 𝜒))
Theorem	bj-nnf-exlim 37199	Proof of the closed form of exlimi 2251 from modalK (compare exlimiv 1949). See also bj-sylget2 37041. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))
Theorem	bj-19.21t 37200	Statement 19.21t 2240 proved from modalK (obsoleting 19.21v 1958). (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))