P. 174 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 17301-17351   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-rals 17301	Define "all some" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴 where 𝜑 is true. An older definition of the "all some" quantifier when scoped to a class, named df-alsc and now removed, instead applied a bare formula 𝜑 to the members of a class, asserting only (∀𝑥 ∈ 𝐴𝜑 ∧ ∃𝑥𝑥 ∈ 𝐴), that is, that the formula held throughout 𝐴 and that 𝐴 had at least one member. I've now decided that that was a mistake. Its older existence conjunct ∃𝑥𝑥 ∈ 𝐴 did not require any member of 𝐴 to satisfy the antecedent, so if the formula was itself an implication, that inner implication could still be vacuously true, which is precisely what the allsome quantifier exists to prevent. For example, the older definition meant that "among Martians, all tall ones are green" could be considered true if there are Martians, but no tall Martians. This version of the definition instead ensures that claims of the form "among Martians, all tall ones are green" can only be true if all tall Martians are green and that there is at least one tall Martian. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	dfrals2 17302	The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
Theorem	alsd 17303	Introduction rule: "all some" holds if the "for all" part holds and the antecedent has a witness. This is the converse of als1d 17305 and als2d 17306 taken together, and is what lets an "all some" statement be proved rather than merely taken apart. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))    &   ⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → ∀∃𝑥(𝜓 → 𝜒))
Theorem	ralsd 17304	Introduction rule for "all some" restricted to a class. This is the converse of rals1d 17307 and rals2d 17308 taken together. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))
Theorem	als1d 17305	Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ (𝜑 → ∀∃𝑥(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
Theorem	als2d 17306	Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ (𝜑 → ∀∃𝑥(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	rals1d 17307	Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
Theorem	rals2d 17308	Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. Note that the witness must satisfy the antecedent 𝜓, not merely be a member of 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	ralsn0d 17309*	Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → 𝐴 ≠ ∅)
Theorem	ralsmd 17310*	Deduction rule: Given "all some" applied to a class, the class is inhabited. This is stronger than ralsn0d 17309, which only concludes that the class is nonempty; see n0r 3535. (Contributed by David A. Wheeler, 20-Jul-2026.)
⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
Theorem	alsex 17311	The consequent of an "all some" is witnessed: if 𝜓 holds of every 𝑥 satisfying 𝜑, and some 𝑥 satisfies 𝜑, then some 𝑥 satisfies 𝜓. This is the positive counterpart of als-no-surprise 17319, and it is the property that ordinary "for all" with implication lacks: from ∀𝑥(𝜑 → 𝜓) alone nothing whatever follows about 𝜓, since nothing need satisfy 𝜑. It is the allsome quantifier says what a speaker of "all Martians are green" usually means. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)
Theorem	ralsex 17312	The consequent of an "all some" restricted to a class is witnessed: some member of 𝐴 satisfying 𝜑 also satisfies 𝜓. Restricted counterpart of alsex 17311. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	alsbii 17313	Congruence: equivalents may be substituted inside an "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑥(𝜒 → 𝜃))
Theorem	ralsbii 17314	Congruence for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥 ∈ 𝐴(𝜒 → 𝜃))
Theorem	alsbid 17315	Deduction form of alsbii 17313. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (∀∃𝑥(𝜓 → 𝜒) ↔ ∀∃𝑥(𝜃 → 𝜏)))
Theorem	nfals 17316	Bound-variable hypothesis builder for "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)
Theorem	nfrals 17317*	Bound-variable hypothesis builder for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃𝑦 ∈ 𝐴(𝜑 → 𝜓)
Theorem	cbvals 17318*	Rule used to change bound variables, using implicit substitution. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑦(𝜒 → 𝜃))
Theorem	als-no-surprise 17319	Demonstrate that there is never a "surprise" when using the allsome quantifier, that is, it is never possible for the consequent to be both always true and always false. This uses the definition of df-als 17300: the universal parts give ∀𝑥¬ 𝜑, which contradicts the witness that the allsome quantifier supplies. Ordinary "for all" with implication has no such property, since ∀𝑥(𝜑 → 𝜓) and ∀𝑥(𝜑 → ¬ 𝜓) can both hold when nothing satisfies 𝜑. (Contributed by David A. Wheeler, 27-Oct-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
⊢ ¬ (∀∃𝑥(𝜑 → 𝜓) ∧ ∀∃𝑥(𝜑 → ¬ 𝜓))
Theorem	rals-no-surprise 17320	Demonstrate that there is never a "surprise" when using the allsome quantifier restricted to a class, that is, it is never possible for the consequent to be both always true and always false of the members of 𝐴 that satisfy the antecedent. This is the restricted counterpart of als-no-surprise 17319, and follows from it by dfrals2 17302. Note that this holds without any assumption that 𝐴 is inhabited; that is the point of allsome, since the corresponding claim for the ordinary restricted "for all" fails when nothing in 𝐴 satisfies 𝜑. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ ¬ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓))
Theorem	ralrals 17321	If the universal part of a restricted "all some" statement holds, then the statement reduces to the existence of a member of 𝐴 satisfying its antecedent. This is the restricted counterpart of ralals 17327. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	rexrals 17322	If a member of 𝐴 satisfying the antecedent exists, then a restricted "all some" statement reduces to its universal part. This is the restricted counterpart of rexals 17328. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)))
Theorem	alsanmo 17323	An "all some" statement conjoined with the claim that at most one 𝑥 satisfies its antecedent is equivalent to the universal part conjoined with the claim that exactly one 𝑥 satisfies the antecedent. The "all some" quantifier supplies the existence of such an 𝑥 and ∃*𝑥𝜑 supplies the at-most-one part, so together they yield ∃!𝑥𝜑. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ ((∀∃𝑥(𝜑 → 𝜓) ∧ ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
Theorem	ralsanmo 17324	An "all some" statement restricted to a class, conjoined with the claim that at most one 𝑥 in 𝐴 satisfies its antecedent, is equivalent to the universal part conjoined with the claim that exactly one 𝑥 in 𝐴 satisfies the antecedent. This is the restricted counterpart of alsanmo 17323. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ ((∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
Theorem	alsralrex 17325*	The general "all some" quantifier with class membership as its antecedent holds if and only if 𝜑 holds for every 𝑥 in 𝐴 and some 𝑥 in 𝐴 satisfies 𝜑. (Contributed by Peter Mazsa, 27-Nov-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	alsraln0m 17326*	The general "all some" quantifier with class membership as its antecedent holds if and only if 𝜑 holds for every 𝑥 in 𝐴 and 𝐴 is inhabited. This is the intuitionistic form of what set.mm states using 𝐴 ≠ ∅; see the section comment. (Contributed by Peter Mazsa, 28-Nov-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))
Theorem	ralals 17327*	If 𝜑 holds for every 𝑥 in 𝐴, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that some 𝑥 in 𝐴 satisfies 𝜑. See ralrals 17321 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	rexals 17328*	If some 𝑥 in 𝐴 satisfies 𝜑, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that 𝜑 holds for every 𝑥 in 𝐴. See rexrals 17322 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	n0alsm 17329*	If 𝐴 is inhabited, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that 𝜑 holds for every 𝑥 in 𝐴. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	2alsraln0m 17330*	Nested general "all some" quantifiers with class membership as their antecedents: 𝜑 holds for every 𝑥 in 𝐴 and every 𝑦 in 𝐵, and both 𝐴 and 𝐵 are inhabited. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 20-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)))
Theorem	2alsraln0idm 17331*	Nested general "all some" quantifiers with class membership as their antecedents, for the same class 𝐴: 𝜑 holds for every 𝑥 and every 𝑦 in 𝐴, and 𝐴 is inhabited. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 20-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))
14.6.3  Allsome one quantifier These are definitions and proofs involving the "allsome one" quantifier, which extends the "allsome" quantifier of the previous section in the same way that ∃! (df-eu 2089) extends ∃. Some systems extend "there exists" by appending a character to it. If a system provides such an extension, it should provide it for allsome as well: append the same character, let it modify allsome's existence conjunct, and change nothing else. Appending "!" gives "allsome one", so ∀∃!𝑥(𝜑 → 𝜓) means that 𝜓 is true whenever 𝜑 is true and that exactly one 𝑥 satisfies 𝜑. This is what the English word "the" usually does. "The king is hungry" claims that a king exists, that there is only one, and that he is hungry, and the form ∀∃!𝑥(𝜑 → 𝜓) claims exactly that kind of statement (specifically when 𝜑 means 𝑥 is a king and 𝜓 means 𝑥 is hungry). English says all of that in a single phrase, and the first two have a dedicated word ("the") for the construct. Many other languages have their own dedicated way of saying this. Languages reserve that kind of compression for what their speakers need constantly, which is a good reason for a formal notation to be able to say it just as directly, rather than spelling it out afresh as a conjunction of two quantified formulas every time it comes up. Russell analyzed such definite descriptions with this apparatus in "On Denoting", where his example of a phrase that appears to refer to someone but in fact denotes no one at all is "the present King of France", item (1) of [Russell1905] p. 479, France being a republic that has no king. Write "the king is hungry" as ∀𝑥(𝜑 → 𝜓) instead and only the last of those three claims survives. The existence claim is silently gone, since that formula is vacuously true when there is no king, and the uniqueness claim is silently gone as well, since that formula holds just the same when there are five kings. Russell reached the opposite verdict for the same example, remarking that every proposition of the form "the present King of France ..." is false, [Russell1905] p. 482. The reason to care is the reason that motivates allsome, only more so. Note that this is not merely a way of writing ∃!𝑥(𝜑 ∧ 𝜓). Reading 𝜑 as "is a king" and 𝜓 as "is hungry", ∃!𝑥(𝜑 ∧ 𝜓) says that there is exactly one hungry king, whereas ∀∃!𝑥(𝜑 → 𝜓) says that there is exactly one king and that he is hungry. The first is true in a region with five kings exactly one of whom is hungry; the second is false there. Uniqueness attaches to the antecedent, not to the conjunction. See dfalseu2 17349 for the exact relationship between the two and alseueu 17350 for the one direction that does hold. Naming: "alseu" is allsome ("als", as in df-als 17300) extended with "exactly one" ("eu", as in df-eu 2089). The form restricted to a class is prefixed with "r", following df-rals 17301 and df-reu 2535, giving df-ralseu 17335. This database is intuitionistic, but nothing in this section depends on excluded middle, so every statement here has the same form as its counterpart in set.mm. That is unlike the allsome section above, where results that recover a witness from 𝐴 ≠ ∅ are omitted or restated in terms of inhabitedness. Soundness: df-alseu 17334 and df-ralseu 17335 are eliminable and conservative directly, so neither needs a justification theorem. Definitions are required to be eliminable and conservative; see the section comment for df-bi 117. Each is a biconditional whose left side is a new syntax construct (walseu 17332 or wralseu 17333) applied to distinct metavariables, and whose right side uses only constructs introduced earlier (∀, ∃!, ∧, →, and the restricted quantifiers df-ral 2533 and df-reu 2535), so any occurrence of the new construct can be replaced by the right side, which is eliminability. Conservativity follows, since a proof of a statement not mentioning ∀∃! can have every use of the definition replaced in this way. Every variable occurring on the right side already occurs on the left side, so no dummy variable is introduced, and introducing a dummy variable whose choice must be shown not to matter is the only circumstance here that would call for a justification theorem. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html 2535
Syntax	walseu 17332	Extend wff definition to include "all some one" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and exactly one 𝑥 satisfies 𝜑. (Contributed by David A. Wheeler, 22-Jul-2026.)
wff ∀∃!𝑥(𝜑 → 𝜓)
Syntax	wralseu 17333	Extend wff definition to include "all some one" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and exactly one 𝑥 in 𝐴 satisfies 𝜑. (Contributed by David A. Wheeler, 22-Jul-2026.)
wff ∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓)
Definition	df-alseu 17334	Define "all some one" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true and exactly one 𝑥 satisfies 𝜑. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
Definition	df-ralseu 17335	Define "all some one" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and exactly one 𝑥 in 𝐴 satisfies 𝜑. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
Theorem	dfralseu2 17336	The bounded "all some one" form is the general form with the class membership folded into the antecedent. This is the "all some one" counterpart of dfrals2 17302. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
Theorem	alseuals 17337	"All some one" implies "all some": requiring exactly one witness is stronger than requiring at least one. Any consequence of an allsome statement is therefore a consequence of the corresponding "all some one" statement, which is how alseu-no-surprise 17351 is proved. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (∀∃!𝑥(𝜑 → 𝜓) → ∀∃𝑥(𝜑 → 𝜓))
Theorem	ralseurals 17338	"All some one" restricted to a class implies "all some" restricted to that class. Restricted counterpart of alseuals 17337. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓))
Theorem	alseud 17339	Introduction rule: "all some one" holds if the "for all" part holds and the antecedent has exactly one witness. This is the converse of alseu1d 17341 and alseu2d 17342 taken together. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))    &   ⊢ (𝜑 → ∃!𝑥𝜓)    ⇒   ⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))
Theorem	ralseud 17340	Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d 17343 and ralseu2d 17344 taken together. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))    &   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))
Theorem	alseu1d 17341	Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
Theorem	alseu2d 17342	Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "exactly one" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∃!𝑥𝜓)
Theorem	ralseu1d 17343	Deduction rule: Given "all some one" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
Theorem	ralseu2d 17344	Deduction rule: Given "all some one" applied to a class, you can extract the "exactly one" part. Note that the witness must satisfy the antecedent 𝜓, not merely be a member of 𝐴. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)
Theorem	alseubii 17345	Congruence: equivalents may be substituted inside an "all some one". This is the "all some one" counterpart of alsbii 17313. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ ∀∃!𝑥(𝜒 → 𝜃))
Theorem	ralseubii 17346	Congruence for "all some one" restricted to a class. This is the "all some one" counterpart of ralsbii 17314. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥 ∈ 𝐴(𝜒 → 𝜃))
Theorem	nfalseu 17347	Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 17316. Unlike the set.mm version of this theorem, no disjoint variable condition is needed, because nfeu 2105 here does not require one. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃!𝑦(𝜑 → 𝜓)
Theorem	nfralseu 17348*	Bound-variable hypothesis builder for "all some one" restricted to a class. This is the "all some one" counterpart of nfrals 17317. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃!𝑦 ∈ 𝐴(𝜑 → 𝜓)
Theorem	dfalseu2 17349	An "all some one" statement is equivalent to its universal part conjoined with the claim that exactly one 𝑥 satisfies both 𝜑 and 𝜓. In other words, given ∀𝑥(𝜑 → 𝜓), requiring exactly one 𝑥 to satisfy 𝜑, which is what df-alseu 17334 requires, and requiring exactly one 𝑥 to satisfy (𝜑 ∧ 𝜓) come to the same thing. Read 𝜑 as "is a king" and 𝜓 as "is hungry": if every king is hungry, then "there is exactly one king" and "there is exactly one hungry king" say the same thing, so either of them, together with "every king is hungry", gives "the king is hungry". The universal conjunct is what makes that work, and it cannot be dropped. ∃!𝑥(𝜑 ∧ 𝜓) on its own is strictly weaker than ∀∃!𝑥(𝜑 → 𝜓), since it is satisfied when many things are 𝜑 and just one of those is 𝜓, as in a region with five kings exactly one of whom is hungry; see alseueu 17350 for the one direction that does hold without it. Uniqueness attaches to the antecedent, not to the conjunction. Russell's analysis of a definite description is built the same way: its uniqueness clause constrains the description predicate alone, while the predication is a separate conjunct. See his worked example of "the father of Charles II was executed", [Russell1905] p. 482. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥(𝜑 ∧ 𝜓)))
Theorem	alseueu 17350	"The 𝜑 is 𝜓 " implies that exactly one thing is both 𝜑 and 𝜓. This is the half of dfalseu2 17349 that drops the universal conjunct; it does not reverse, so ∃!𝑥(𝜑 ∧ 𝜓) cannot be used in place of an "all some one" statement. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ (∀∃!𝑥(𝜑 → 𝜓) → ∃!𝑥(𝜑 ∧ 𝜓))
Theorem	alseu-no-surprise 17351	Demonstrate that there is never a "surprise" when using the "all some one" quantifier, that is, it is never possible for the consequent to be both always true and always false. This follows from als-no-surprise 17319 by alseuals 17337. See als-no-surprise 17319 for why ordinary "for all" with implication has no such property. (Contributed by David A. Wheeler, 22-Jul-2026.)
⊢ ¬ (∀∃!𝑥(𝜑 → 𝜓) ∧ ∀∃!𝑥(𝜑 → ¬ 𝜓))

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