P. 373 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37201-37300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-19.41t 37201	Closed form of 19.41 2269 from the same axioms as 19.41v 1968. The same is doable with 19.27 2261, 19.28 2262, 19.31 2268, 19.32 2267, 19.44 2271, 19.45 2272. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)))
Theorem	bj-pm11.53vw 37202	Version of pm11.53v 1963 with nonfreeness antecedents. One can also prove the theorem with antecedent (Ⅎ'𝑦∀𝑥𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓). (Contributed by BJ, 7-Oct-2024.)
⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-nnfv 37203*	A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.)
⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnfbd 37204*	If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi 37182. (Contributed by BJ, 27-Aug-2023.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
Theorem	bj-pm11.53a 37205*	A variant of pm11.53v 1963. One can similarly prove a variant with DV (𝑦, 𝜑) and ∀𝑦Ⅎ'𝑥𝜓 instead of DV (𝑥, 𝜓) and ∀𝑥Ⅎ'𝑦𝜑. (Contributed by BJ, 7-Oct-2024.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-equsvt 37206*	A variant of equsv 2022. (Contributed by BJ, 7-Oct-2024.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	bj-equsalvwd 37207*	Variant of equsalvw 2023. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
Theorem	bj-equsexvwd 37208*	Variant of equsexvw 2024. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒))
Theorem	bj-nnf-spim 37209*	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 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-nnf-spime 37210*	An existential generalization result in deduction form, 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 → Ⅎ'𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∃𝑥𝜒))
Theorem	bj-nnf-cbvaliv 37211*	The only DV conditions are those saying that 𝑦 is a fresh variable used to construct 𝜒. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	bj-sbievwd 37212*	Variant of sbievw 2126. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	bj-sbft 37213	Version of sbft 2303 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.)
⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑))
Theorem	bj-nnf-cbvali 37214*	Compared with bj-nnf-cbvaliv 37211, replacing the DV condition on 𝑦, 𝜓 with the nonfreeness condition requires ax-11 2190. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	bj-nnf-cbval 37215*	Compared with cbvalv1 2371, this saves ax-12 2211. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-dfnnf3 37216	Alternate definition of nonfreeness when sp 2217 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1803. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nfnnfTEMP 37217	New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2217. (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-wnfnf 37218	When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). Note that this could also be proved from bj-nnfim 37187, bj-nnfe1 37220 and bj-nnfa1 37219. (Contributed by BJ, 9-Dec-2023.)
⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
Theorem	bj-nnfa1 37219	See nfa1 2184. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ Ⅎ'𝑥∀𝑥𝜑
Theorem	bj-nnfe1 37220	See nfe1 2183. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ Ⅎ'𝑥∃𝑥𝜑
Theorem	bj-nnflemaa 37221	One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 37115. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜓))
Theorem	bj-nnflemee 37222	One of four lemmas for nonfreeness: antecedent and consequent both expressed using existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-nnflemae 37223	One of four lemmas for nonfreeness: antecedent expressed with universal quantifier and consequent expressed with existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜓))
Theorem	bj-nnflemea 37224	One of four lemmas for nonfreeness: antecedent expressed with existential quantifier and consequent expressed with universal quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-nnfalt 37225	See nfal 2354 and bj-nfalt 37148. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∀𝑥𝜑)
Theorem	bj-nnfext 37226	See nfex 2355 and bj-nfext 37149. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∃𝑥𝜑)
Theorem	bj-pm11.53v 37227	Version of pm11.53v 1963 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.)
⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
21.19.4.13  Adding ax-13
Theorem	bj-axc10 37228	Alternate proof of axc10 2415. Shorter. One can prove a version with DV (𝑥, 𝑦) without ax-13 2402, by using ax6ev 1988 instead of ax6e 2413. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-alequex 37229	A fol lemma. See alequexv 2020 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2416 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
Theorem	bj-spimt2 37230	A step in the proof of spimt 2416. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-cbv3ta 37231	Closed form of cbv3 2427. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-cbv3tb 37232	Closed form of cbv3 2427. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-hbsb3t 37233	A theorem close to a closed form of hbsb3 2517. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
Theorem	bj-hbsb3 37234	Shorter proof of hbsb3 2517. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1t 37235	A theorem close to a closed form of nfs1 2518. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1t2 37236	A theorem close to a closed form of nfs1 2518. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1 37237	Shorter proof of nfs1 2518 (three essential steps instead of four). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
21.19.4.14  Removing dependencies on ax-13 (and ax-11) It is known that ax-13 2402 is logically redundant (see ax13w 2169 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2402 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 2402 with ax13w 2169. This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2402 (and using ax6v 1987 / ax6ev 1988 instead of ax-6 1986 / ax6e 2413, as is currently done). One reason to be optimistic is that the first few utility theorems using ax-13 2402 (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 2402, labeled bj-xxxv (we follow the proof of xxx but use ax6v 1987 and ax6ev 1988 instead of ax-6 1986 and ax6e 2413, and ax-5 1929 instead of ax13v 2403; 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 dispense with ax-13 2402, so as to remove dependencies on ax-13 2402 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 2190, typically by replacing a nonfree hypothesis with a disjoint variable condition (see cbv3v2 2275 and following theorems).
Theorem	bj-axc10v 37238*	Version of axc10 2415 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-spimtv 37239*	Version of spimt 2416 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbv3hv2 37240*	Version of cbv3h 2434 with two disjoint variable conditions, which does not require ax-11 2190 nor ax-13 2402. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	bj-cbv1hv 37241*	Version of cbv1h 2435 with a disjoint variable condition, which does not require ax-13 2402. (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 37242*	Version of cbv2h 2436 with a disjoint variable condition, which does not require ax-13 2402. (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 37243*	Version of cbv2 2433 with a disjoint variable condition, which does not require ax-13 2402. (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 37244*	Version of cbvald 2437 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvexdv 37245*	Version of cbvexd 2438 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	bj-cbval2vv 37246*	Version of cbval2vv 2443 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
Theorem	bj-cbvex2vv 37247*	Version of cbvex2vv 2444 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
Theorem	bj-cbvaldvav 37248*	Version of cbvaldva 2439 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvexdvav 37249*	Version of cbvexdva 2440 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	bj-cbvex4vv 37250*	Version of cbvex4v 2445 with a disjoint variable condition, which does not require ax-13 2402. (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 37251*	Version of equsalh 2450 with a disjoint variable condition, which does not require ax-13 2402. Remark: this is the same as equsalhw 2324. TODO: delete after moving the following paragraph somewhere. Remarks: equsexvw 2024 has been moved to Main; Theorem ax13lem2 2406 has a DV version which is a simple consequence of ax5e 1931; Theorems nfeqf2 2407, dveeq2 2408, nfeqf1 2409, dveeq1 2410, nfeqf 2411, axc9 2412, ax13 2405, have dv versions which are simple consequences of ax-5 1929. (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 37252*	Version of axc11n 2456 with a disjoint variable condition; instance of aevlem 2076. 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 37253*	Version of aecoms 2458 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2459 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5401). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	bj-axc11v 37254*	Version of axc11 2460 with a disjoint variable condition, which does not require ax-13 2402 nor ax-10 2174. Remark: the following theorems (hbae 2461, nfae 2463, hbnae 2462, nfnae 2464, hbnaes 2465) would need to be totally unbundled to be proved without ax-13 2402, hence would be simple consequences of ax-5 1929 or nfv 1933. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Theorem	bj-drnf2v 37255*	Version of drnf2 2474 with a disjoint variable condition, which does not require ax-10 2174, ax-11 2190, ax-12 2211, ax-13 2402. Instance of nfbidv 1941. Note that the version of axc15 2452 with a disjoint variable condition is actually ax12v2 2213 (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 37256*	Version of equs45f 2489 with a disjoint variable condition, which does not require ax-13 2402. Note that the version of equs5 2490 with a disjoint variable condition is actually sbalex 2276 (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 37257*	Version of hbsb2 2512 with a disjoint variable condition, which does not require ax-13 2402, and removal of ax-13 2402 from hbs1 2307. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1v 37258*	Version of nfsb2 2513 with a disjoint variable condition, which does not require ax-13 2402, and removal of ax-13 2402 from nfs1v 2189. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
Theorem	bj-hbsb2av 37259*	Version of hbsb2a 2514 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-hbsb3v 37260*	Version of hbsb3 2517 with a disjoint variable condition, which does not require ax-13 2402. (Remark: the unbundled version of nfs1 2518 is given by bj-nfs1v 37258.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfsab1 37261*	Remove dependency on ax-13 2402 from nfsab1 2747. UPDATE / TODO: nfsab1 2747 does not use ax-13 2402 either anymore; bj-nfsab1 37261 is shorter than nfsab1 2747 but uses ax-12 2211. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
Theorem	bj-dtrucor2v 37262*	Version of dtrucor2 5326 with a disjoint variable condition, which does not require ax-13 2402 (nor ax-4 1828, ax-5 1929, ax-7 2027, ax-12 2211). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)    ⇒   ⊢ (𝜑 ∧ ¬ 𝜑)
21.19.4.15  Distinct var metavariables 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 37263	Biconditional version of a form of hbae 2461 with commuted quantifiers, not requiring ax-11 2190. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦)
Theorem	bj-hbaeb 37264	Biconditional version of hbae 2461. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)
Theorem	bj-hbnaeb 37265	Biconditional version of hbnae 2462 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	bj-dvv 37266	A special instance of bj-hbaeb2 37263. 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)
21.19.4.16  Around ~ equsal 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 36993), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2448 (and equsalh 2450 and equsexh 2451). Even if only one of these two theorems holds, it should be added to the database.
Theorem	bj-equsal1t 37267	Duplication of wl-equsal1t 38005, with shorter proof. If one imposes a disjoint variable condition on 𝑥, 𝑦, then one can use alequexv 2020 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 38006 is also interesting. (Contributed by BJ, 6-Oct-2018.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	bj-equsal1ti 37268	Inference associated with bj-equsal1t 37267. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)
Theorem	bj-equsal1 37269	One direction of equsal 2447. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓)
Theorem	bj-equsal2 37270	One direction of equsal 2447. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓))
Theorem	bj-equsal 37271	Shorter proof of equsal 2447. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2447, but "min */exc equsal" is ok. (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
21.19.4.17  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".
Theorem	stdpc5t 37272	Closed form of stdpc5 2242. (Possible to place it before 19.21t 2240 and use it to prove 19.21t 2240). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-stdpc5 37273	More direct proof of stdpc5 2242. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	2stdpc5 37274	A double stdpc5 2242 (one direction of PM*11.3). See also 2stdpc4 2100 and 19.21vv 44912. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-19.21t0 37275	Proof of 19.21t 2240 from stdpc5t 37272. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	exlimii 37276	Inference associated with exlimi 2251. 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 37277	Proof of ax-11 2190 similar to PM's proof of alcom 2192 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 37281. Axiom ax-11 2190 is used in the proof only through nfa2 2208. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	ax6er 37278	Commuted form of ax6e 2413. (Could be placed right after ax6e 2413). (Contributed by BJ, 15-Sep-2018.)
⊢ ∃𝑥 𝑦 = 𝑥
Theorem	exlimiieq1 37279	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 37280	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 37281*	Proof of ax-11 2190 from the standard axioms of predicate calculus, similar to PM's proof of alcom 2192 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 2190 is used in the proof only through nfal 2354, nfsb 2553, sbal 2202, sb8 2547. See also ax11-pm 37277. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
21.19.4.18  Alternate definition of substitution
Theorem	bj-sbsb 37282	Biconditional showing two possible (dual) definitions of substitution df-sb 2090 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 37283	Alternate (dual) definition of substitution df-sb 2090 not using dummy variables. (Contributed by BJ, 19-Mar-2021.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∨ (𝑥 = 𝑦 ∧ 𝜑)))
21.19.4.19  Lemmas for substitution
Theorem	bj-sbf3 37284	Substitution has no effect on a bound variable (existential quantifier case); see sbf2 2305. (Contributed by BJ, 2-May-2019.)
⊢ ([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	bj-sbf4 37285	Substitution has no effect on a bound variable (nonfreeness case); see sbf2 2305. (Contributed by BJ, 2-May-2019.)
⊢ ([𝑦 / 𝑥]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜑)
21.19.4.20  Existential uniqueness
Theorem	bj-eu3f 37286*	Version of eu3v 2596 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 2596. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
21.19.4.21  First-order logic: miscellaneous Miscellaneous theorems of first-order logic.
Theorem	bj-sblem1 37287*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜒)))
Theorem	bj-sblem2 37288*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → ((∃𝑥𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-sblem 37289*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ (∀𝑥(𝜑 → (𝜓 ↔ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜒)))
Theorem	bj-sbievw1 37290*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → 𝜓))
Theorem	bj-sbievw2 37291*	Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
⊢ ([𝑦 / 𝑥](𝜓 → 𝜑) → (𝜓 → [𝑦 / 𝑥]𝜑))
Theorem	bj-sbievw 37292*	Lemma for substitution. Closed form of equsalvw 2023 and sbievw 2126. (Contributed by BJ, 23-Jul-2023.)
⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
Theorem	bj-sbievv 37293	Version of sbie 2532 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 37294	Uniqueness is equivalent to existence being equivalent to unique existence. (Contributed by BJ, 14-Oct-2022.)
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
Theorem	bj-sbidmOLD 37295	Obsolete proof of sbidm 2540 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 37296*	Deduction form of dvelim 2481 with disjoint variable conditions. Uncurried (imported) form of bj-dvelimdv1 37297. 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 1933 can be replaced with nfal 2354 followed by nfn 1876. Remark: nfald 2359 uses ax-11 2190; it might be possible to inline and use ax11w 2163 instead, but there is still a use via 19.12 2358 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 37297*	Curried (exported) form of bj-dvelimdv 37296 (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 37298*	A version of dvelim 2481 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 37299*	Nonfreeness in a membership statement. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧)
Theorem	bj-axc14nf 37300	Proof of a version of axc14 2493 using the "nonfree" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦))