P. 353 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35201-35300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-nnfeai 35201	Nonfreeness implies the equivalent of ax5ea 1916, inference form. (Contributed by BJ, 22-Sep-2024.)
⊢ Ⅎ'𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
Theorem	bj-dfnnf2 35202	Alternate definition of df-bj-nnf 35189 using only primitive symbols (→, ¬, ∀) in each conjunct. (Contributed by BJ, 20-Aug-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ ((𝜑 → ∀𝑥𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
Theorem	bj-nnfnfTEMP 35203	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 1786 except via df-nf 1786 directly. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	bj-wnfnf 35204	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 35211, bj-nnfe1 35225 and bj-nnfa1 35224. (Contributed by BJ, 9-Dec-2023.)
⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
Theorem	bj-nnfnt 35205	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 35211). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1859. (Contributed by BJ, 28-Jul-2023.)
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)
Theorem	bj-nnftht 35206	A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2176 (modal T), as in bj-nnfbi 35190. (Contributed by BJ, 28-Jul-2023.)
⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)
Theorem	bj-nnfth 35207	A variable is nonfree in a theorem, inference form. (Contributed by BJ, 28-Jul-2023.)
⊢ 𝜑    ⇒   ⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnfnth 35208	A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.)
⊢ ¬ 𝜑    ⇒   ⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnfim1 35209	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 35210	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-nnfim 35211	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 35212	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 35213	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 35211, bj-nnfnt 35205 and bj-nnfbi 35190, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∧ 𝜓))
Theorem	bj-nnfand 35214	Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 35213, 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 35213 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 35214 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 35215	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 35211, bj-nnfnt 35205 and bj-nnfbi 35190, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∨ 𝜓))
Theorem	bj-nnford 35216	Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 35215 and bj-nnfand 35214. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
⊢ (𝜑 → Ⅎ'𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒))
Theorem	bj-nnfbit 35217	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 35218	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-nnfv 35219*	A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.)
⊢ Ⅎ'𝑥𝜑
Theorem	bj-nnf-alrim 35220	Proof of the closed form of alrimi 2206 from modalK (compare alrimiv 1930). See also bj-alrim 35158. Actually, most proofs between 19.3t 2194 and 2sbbid 2239 could be proved without ax-12 2171. (Contributed by BJ, 20-Aug-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-nnf-exlim 35221	Proof of the closed form of exlimi 2210 from modalK (compare exlimiv 1933). See also bj-sylget2 35086. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))
Theorem	bj-dfnnf3 35222	Alternate definition of nonfreeness when sp 2176 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1786. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nfnnfTEMP 35223	New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2176. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1786 except via df-nf 1786 directly. (Proof modification is discouraged.)
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)
Theorem	bj-nnfa1 35224	See nfa1 2148. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ Ⅎ'𝑥∀𝑥𝜑
Theorem	bj-nnfe1 35225	See nfe1 2147. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ Ⅎ'𝑥∃𝑥𝜑
Theorem	bj-19.12 35226	See 19.12 2320. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2162 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1786 or df-bj-nnf 35189, directly or indirectly. (Proof modification is discouraged.)
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)
Theorem	bj-nnflemaa 35227	One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 35146. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
Theorem	bj-nnflemee 35228	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 → 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑))
Theorem	bj-nnflemae 35229	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 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
Theorem	bj-nnflemea 35230	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 → 𝜑) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑))
Theorem	bj-nnfalt 35231	See nfal 2316 and bj-nfalt 35176. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∀𝑥𝜑)
Theorem	bj-nnfext 35232	See nfex 2317 and bj-nfext 35177. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∃𝑥𝜑)
Theorem	bj-stdpc5t 35233	Alias of bj-nnf-alrim 35220 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2201 proved from modalK (obsoleting stdpc5v 1941). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 35220 instead. (New usaged is discouraged.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-19.21t 35234	Statement 19.21t 2199 proved from modalK (obsoleting 19.21v 1942). (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	bj-19.23t 35235	Statement 19.23t 2203 proved from modalK (obsoleting 19.23v 1945). (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	bj-19.36im 35236	One direction of 19.36 2223 from the same axioms as 19.36imv 1948. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-19.37im 35237	One direction of 19.37 2225 from the same axioms as 19.37imv 1951. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))
Theorem	bj-19.42t 35238	Closed form of 19.42 2229 from the same axioms as 19.42v 1957. (Contributed by BJ, 2-Dec-2023.)
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))
Theorem	bj-19.41t 35239	Closed form of 19.41 2228 from the same axioms as 19.41v 1953. 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 35240	Version of sbft 2261 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.)
⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑))
Theorem	bj-pm11.53vw 35241	Version of pm11.53v 1947 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-pm11.53v 35242	Version of pm11.53v 1947 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.)
⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-pm11.53a 35243*	A variant of pm11.53v 1947. 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 35244*	A variant of equsv 2006. (Contributed by BJ, 7-Oct-2024.)
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	bj-equsalvwd 35245*	Variant of equsalvw 2007. (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 35246*	Variant of equsexvw 2008. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒))
Theorem	bj-sbievwd 35247*	Variant of sbievw 2095. (Contributed by BJ, 7-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ'𝑥𝜒)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
21.15.4.11  Adding ax-13
Theorem	bj-axc10 35248	Alternate proof of axc10 2383. Shorter. One can prove a version with DV (𝑥, 𝑦) without ax-13 2370, by using ax6ev 1973 instead of ax6e 2381. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-alequex 35249	A fol lemma. See alequexv 2004 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2384 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
Theorem	bj-spimt2 35250	A step in the proof of spimt 2384. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
Theorem	bj-cbv3ta 35251	Closed form of cbv3 2395. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-cbv3tb 35252	Closed form of cbv3 2395. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Theorem	bj-hbsb3t 35253	A theorem close to a closed form of hbsb3 2489. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
Theorem	bj-hbsb3 35254	Shorter proof of hbsb3 2489. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1t 35255	A theorem close to a closed form of nfs1 2490. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1t2 35256	A theorem close to a closed form of nfs1 2490. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1 35257	Shorter proof of nfs1 2490 (three essential steps instead of four). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
21.15.4.12  Removing dependencies on ax-13 (and ax-11) It is known that ax-13 2370 is logically redundant (see ax13w 2132 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2370 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 2370 with ax13w 2132. This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2370 (and using ax6v 1972 / ax6ev 1973 instead of ax-6 1971 / ax6e 2381, as is currently done). One reason to be optimistic is that the first few utility theorems using ax-13 2370 (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 2370, labeled bj-xxxv (we follow the proof of xxx but use ax6v 1972 and ax6ev 1973 instead of ax-6 1971 and ax6e 2381, and ax-5 1913 instead of ax13v 2371; 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 2370, so as to remove dependencies on ax-13 2370 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 2154, typically by replacing a nonfree hypothesis with a disjoint variable condition (see cbv3v2 2234 and following theorems).
Theorem	bj-axc10v 35258*	Version of axc10 2383 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	bj-spimtv 35259*	Version of spimt 2384 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))
Theorem	bj-cbv3hv2 35260*	Version of cbv3h 2402 with two disjoint variable conditions, which does not require ax-11 2154 nor ax-13 2370. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	bj-cbv1hv 35261*	Version of cbv1h 2403 with a disjoint variable condition, which does not require ax-13 2370. (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 35262*	Version of cbv2h 2404 with a disjoint variable condition, which does not require ax-13 2370. (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 35263*	Version of cbv2 2401 with a disjoint variable condition, which does not require ax-13 2370. (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 35264*	Version of cbvald 2405 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvexdv 35265*	Version of cbvexd 2406 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	bj-cbval2vv 35266*	Version of cbval2vv 2411 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
Theorem	bj-cbvex2vv 35267*	Version of cbvex2vv 2412 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
Theorem	bj-cbvaldvav 35268*	Version of cbvaldva 2407 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	bj-cbvexdvav 35269*	Version of cbvexdva 2408 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	bj-cbvex4vv 35270*	Version of cbvex4v 2413 with a disjoint variable condition, which does not require ax-13 2370. (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 35271*	Version of equsalh 2418 with a disjoint variable condition, which does not require ax-13 2370. Remark: this is the same as equsalhw 2287. TODO: delete after moving the following paragraph somewhere. Remarks: equsexvw 2008 has been moved to Main; Theorem ax13lem2 2374 has a DV version which is a simple consequence of ax5e 1915; Theorems nfeqf2 2375, dveeq2 2376, nfeqf1 2377, dveeq1 2378, nfeqf 2379, axc9 2380, ax13 2373, have dv versions which are simple consequences of ax-5 1913. (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 35272*	Version of axc11n 2424 with a disjoint variable condition; instance of aevlem 2058. 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 35273*	Version of aecoms 2426 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2427 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5393). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	bj-axc11v 35274*	Version of axc11 2428 with a disjoint variable condition, which does not require ax-13 2370 nor ax-10 2137. Remark: the following theorems (hbae 2429, nfae 2431, hbnae 2430, nfnae 2432, hbnaes 2433) would need to be totally unbundled to be proved without ax-13 2370, hence would be simple consequences of ax-5 1913 or nfv 1917. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Theorem	bj-drnf2v 35275*	Version of drnf2 2442 with a disjoint variable condition, which does not require ax-10 2137, ax-11 2154, ax-12 2171, ax-13 2370. Instance of nfbidv 1925. Note that the version of axc15 2420 with a disjoint variable condition is actually ax12v2 2173 (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 35276*	Version of equs45f 2457 with a disjoint variable condition, which does not require ax-13 2370. Note that the version of equs5 2458 with a disjoint variable condition is actually sbalex 2235 (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 35277*	Version of hbsb2 2484 with a disjoint variable condition, which does not require ax-13 2370, and removal of ax-13 2370 from hbs1 2265. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfs1v 35278*	Version of nfsb2 2485 with a disjoint variable condition, which does not require ax-13 2370, and removal of ax-13 2370 from nfs1v 2153. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
Theorem	bj-hbsb2av 35279*	Version of hbsb2a 2486 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-hbsb3v 35280*	Version of hbsb3 2489 with a disjoint variable condition, which does not require ax-13 2370. (Remark: the unbundled version of nfs1 2490 is given by bj-nfs1v 35278.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	bj-nfsab1 35281*	Remove dependency on ax-13 2370 from nfsab1 2721. UPDATE / TODO: nfsab1 2721 does not use ax-13 2370 either anymore; bj-nfsab1 35281 is shorter than nfsab1 2721 but uses ax-12 2171. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
Theorem	bj-dtrucor2v 35282*	Version of dtrucor2 5327 with a disjoint variable condition, which does not require ax-13 2370 (nor ax-4 1811, ax-5 1913, ax-7 2011, ax-12 2171). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)    ⇒   ⊢ (𝜑 ∧ ¬ 𝜑)
21.15.4.13  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 35283	Biconditional version of a form of hbae 2429 with commuted quantifiers, not requiring ax-11 2154. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦)
Theorem	bj-hbaeb 35284	Biconditional version of hbae 2429. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦)
Theorem	bj-hbnaeb 35285	Biconditional version of hbnae 2430 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	bj-dvv 35286	A special instance of bj-hbaeb2 35283. 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)
21.15.4.14  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 35055), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2416 (and equsalh 2418 and equsexh 2419). Even if only one of these two theorems holds, it should be added to the database.
Theorem	bj-equsal1t 35287	Duplication of wl-equsal1t 36000, with shorter proof. If one imposes a disjoint variable condition on x,y , then one can use alequexv 2004 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 36001 is also interesting. (Contributed by BJ, 6-Oct-2018.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	bj-equsal1ti 35288	Inference associated with bj-equsal1t 35287. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)
Theorem	bj-equsal1 35289	One direction of equsal 2415. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓)
Theorem	bj-equsal2 35290	One direction of equsal 2415. (Contributed by BJ, 30-Sep-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓))
Theorem	bj-equsal 35291	Shorter proof of equsal 2415. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2415, but "min */exc equsal" is ok. (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
21.15.4.15  Some Principia Mathematica proofs References are made to the second edition (1927, reprinted 1963) of Principia Mathematica, Vol. 1. Theorems are referred to in the form "PM*xx.xx".
Theorem	stdpc5t 35292	Closed form of stdpc5 2201. (Possible to place it before 19.21t 2199 and use it to prove 19.21t 2199). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
Theorem	bj-stdpc5 35293	More direct proof of stdpc5 2201. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	2stdpc5 35294	A double stdpc5 2201 (one direction of PM*11.3). See also 2stdpc4 2073 and 19.21vv 42646. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-19.21t0 35295	Proof of 19.21t 2199 from stdpc5t 35292. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	exlimii 35296	Inference associated with exlimi 2210. 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 35297	Proof of ax-11 2154 similar to PM's proof of alcom 2156 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 35301. Axiom ax-11 2154 is used in the proof only through nfa2 2170. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	ax6er 35298	Commuted form of ax6e 2381. (Could be placed right after ax6e 2381). (Contributed by BJ, 15-Sep-2018.)
⊢ ∃𝑥 𝑦 = 𝑥
Theorem	exlimiieq1 35299	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 35300	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