P. 507 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50601-50680   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rals1d 50601	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 50602	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 50603*	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	alsex 50604	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 50612, and it is the property that ordinary "for all" with implication lacks: from ∀𝑥(𝜑 → 𝜓) alone nothing whatever follows about 𝜓, as alimp-surprise 50586 shows. It is the reason 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 50605	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 50604. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	alsbii 50606	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 50607	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 50608	Deduction form of alsbii 50606. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (∀∃𝑥(𝜓 → 𝜒) ↔ ∀∃𝑥(𝜃 → 𝜏)))
Theorem	nfals 50609	Bound-variable hypothesis builder for "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)
Theorem	nfrals 50610*	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 50611*	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 50612	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 50594; the proof itself builds on alimp-no-surprise 50587. For a contrast, see alimp-surprise 50586. (Contributed by David A. Wheeler, 27-Oct-2018.)
⊢ ¬ (∀∃𝑥(𝜑 → 𝜓) ∧ ∀∃𝑥(𝜑 → ¬ 𝜓))
Theorem	rals-no-surprise 50613	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 50612, and follows from it by dfrals2 50596. Note that this needs no assumption that 𝐴 is nonempty, because allsome requires a member of 𝐴 satisfying 𝜑, and that member would have to satisfy both 𝜓 and ¬ 𝜓. The ordinary restricted "for all" requires no such member and can be vacuously true, as shown in empty-surprise2 50589; that is the point of allsome. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ ¬ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓))
Theorem	ralrals 50614	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 50620. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	rexrals 50615	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 50621. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)))
Theorem	alsanmo 50616	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 50617	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 50616. (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 50618*	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, 15-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	alsraln0 50619*	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 not empty. (Contributed by Peter Mazsa, 28-Nov-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))
Theorem	ralals 50620*	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 50614 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 50621*	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 50615 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	n0als 50622*	If 𝐴 is not empty, 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, 15-Jul-2026.)
⊢ (𝐴 ≠ ∅ → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	2alsraln0 50623*	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 not empty. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)))
Theorem	2alsraln0id 50624*	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 not empty. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))
21.52.13  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 2597) 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; see alimp-surprise 50586 and empty-surprise 50588. 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 50642 for the exact relationship between the two and alseueu 50643 for the one direction that does hold. Naming: "alseu" is allsome ("als", as in df-als 50594) extended with "exactly one" ("eu", as in df-eu 2597). The form restricted to a class is prefixed with "r", following df-rals 50595 and df-reu 3370, giving df-ralseu 50628. Soundness: df-alseu 50627 and df-ralseu 50628 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 210. Each is a biconditional whose left side is a new syntax construct (walseu 50625 or wralseu 50626) applied to distinct metavariables, and whose right side uses only constructs introduced earlier (∀, ∃!, ∧, →, and the restricted quantifiers df-ral 3080 and df-reu 3370), 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 3370
Syntax	walseu 50625	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, 21-Jul-2026.)
wff ∀∃!𝑥(𝜑 → 𝜓)
Syntax	wralseu 50626	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, 21-Jul-2026.)
wff ∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓)
Definition	df-alseu 50627	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, 21-Jul-2026.)
⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
Definition	df-ralseu 50628	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, 21-Jul-2026.)
⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
Theorem	dfralseu2 50629	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 50596. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
Theorem	alseuals 50630	"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 50644 is proved. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (∀∃!𝑥(𝜑 → 𝜓) → ∀∃𝑥(𝜑 → 𝜓))
Theorem	ralseurals 50631	"All some one" restricted to a class implies "all some" restricted to that class. Restricted counterpart of alseuals 50630. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓))
Theorem	alseud 50632	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 50634 and alseu2d 50635 taken together. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))    &   ⊢ (𝜑 → ∃!𝑥𝜓)    ⇒   ⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))
Theorem	ralseud 50633	Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d 50636 and ralseu2d 50637 taken together. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))    &   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))
Theorem	alseu1d 50634	Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
Theorem	alseu2d 50635	Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "exactly one" part. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∃!𝑥𝜓)
Theorem	ralseu1d 50636	Deduction rule: Given "all some one" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
Theorem	ralseu2d 50637	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, 21-Jul-2026.)
⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)
Theorem	alseubii 50638	Congruence: equivalents may be substituted inside an "all some one". This is the "all some one" counterpart of alsbii 50606. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ ∀∃!𝑥(𝜒 → 𝜃))
Theorem	ralseubii 50639	Congruence for "all some one" restricted to a class. This is the "all some one" counterpart of ralsbii 50607. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥 ∈ 𝐴(𝜒 → 𝜃))
Theorem	nfalseu 50640*	Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 50609. Unlike nfals 50609 it requires 𝑥 and 𝑦 to be disjoint, because the corresponding builder for ∃! is nfeuw 2621, which requires it; the version without that requirement, nfeu 2622, depends on ax-13 2404 and its use is discouraged. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃!𝑦(𝜑 → 𝜓)
Theorem	nfralseu 50641*	Bound-variable hypothesis builder for "all some one" restricted to a class. This is the "all some one" counterpart of nfrals 50610. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃!𝑦 ∈ 𝐴(𝜑 → 𝜓)
Theorem	dfalseu2 50642	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 50627 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 50643 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, 21-Jul-2026.)
⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥(𝜑 ∧ 𝜓)))
Theorem	alseueu 50643	"The 𝜑 is 𝜓 " implies that exactly one thing is both 𝜑 and 𝜓. This is the half of dfalseu2 50642 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, 21-Jul-2026.)
⊢ (∀∃!𝑥(𝜑 → 𝜓) → ∃!𝑥(𝜑 ∧ 𝜓))
Theorem	alseu-no-surprise 50644	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 50612 by alseuals 50630. For a contrast, see alimp-surprise 50586. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ ¬ (∀∃!𝑥(𝜑 → 𝜓) ∧ ∀∃!𝑥(𝜑 → ¬ 𝜓))
21.52.14  Miscellaneous Miscellaneous proofs.
Theorem	5m4e1 50645	Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
⊢ (5 − 4) = 1
Theorem	2p2ne5 50646	Prove that 2 + 2 ≠ 5. In George Orwell's "1984", Part One, Chapter Seven, the protagonist Winston notes that, "In the end the Party would announce that two and two made five, and you would have to believe it." http://www.sparknotes.com/lit/1984/section4.rhtml. More generally, the phrase 2 + 2 = 5 has come to represent an obviously false dogma one may be required to believe. See the Wikipedia article for more about this: https://en.wikipedia.org/wiki/2_%2B_2_%3D_5. Unsurprisingly, we can easily prove that this claim is false. (Contributed by David A. Wheeler, 31-Jan-2017.)
⊢ (2 + 2) ≠ 5
Theorem	resolution 50647	Resolution rule. This is the primary inference rule in some automated theorem provers such as prover9. The resolution rule can be traced back to Davis and Putnam (1960). (Contributed by David A. Wheeler, 9-Feb-2017.)
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → (𝜓 ∨ 𝜒))
Theorem	testable 50648	In classical logic all wffs are testable, that is, it is always true that (¬ 𝜑 ∨ ¬ ¬ 𝜑). This is not necessarily true in intuitionistic logic. In intuitionistic logic, if this statement is true for some 𝜑, then 𝜑 is testable. The proof is trivial because it's simply a special case of the law of the excluded middle, which is true in classical logic but not necessarily true in intuitionisic logic. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑)
21.52.15  Theorems about algebraic numbers
Theorem	aacllem 50649*	Lemma for other theorems about 𝔸. (Contributed by Brendan Leahy, 3-Jan-2020.) (Revised by Alexander van der Vekens and David A. Wheeler, 25-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑𝑘) = Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝔸)
21.53  Mathbox for Jiamin Zhao
21.53.1  Cross product and scalar triple product in RR^3
Theorem	1ne3 50650	1 is not equal to 3. (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 1 ≠ 3
Theorem	2ne3 50651	2 is not equal to 3. (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 2 ≠ 3
Theorem	1elfz13 50652	Membership of 1 in the integer interval ( 1 ... 3 ). (Suggested by avekens.) (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 1 ∈ (1...3)
Theorem	2elfz13 50653	Membership of 2 in the integer interval ( 1 ... 3 ). (Suggested by tirix.) (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 2 ∈ (1...3)
Theorem	3elfz13 50654	Membership of 3 in the integer interval ( 1 ... 3 ). (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 3 ∈ (1...3)
Theorem	rr3fvcl 50655	The components of a 3-dimensional real coordinate vector are real numbers. (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ (𝐴 ∈ (ℝ ↑m (1...3)) → ((𝐴‘1) ∈ ℝ ∧ (𝐴‘2) ∈ ℝ ∧ (𝐴‘3) ∈ ℝ))
Theorem	rr3fv1cli 50656	First component of a 3-dimensional real coordinate vector is real. (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (𝐴‘1) ∈ ℝ
Theorem	rr3fv2cli 50657	Second component of a 3-dimensional real coordinate vector is real. (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (𝐴‘2) ∈ ℝ
Theorem	rr3fv3cli 50658	Third component of a 3-dimensional real coordinate vector is real. (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (𝐴‘3) ∈ ℝ
Syntax	ccrossp 50659	Extend class notation to include the cross product operation. (Contributed by Jiamin Zhao, 31-Jul-2026.)
class ⊠
Definition	df-crossp 50660*	Define the cross product of two 3-dimensional real coordinate vectors. Vectors are represented as functions on (1...3). (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ ⊠ = (𝑢 ∈ (ℝ ↑m (1...3)), 𝑣 ∈ (ℝ ↑m (1...3)) ↦ (𝑘 ∈ (1...3) ↦ if(𝑘 = 1, (((𝑢‘2) · (𝑣‘3)) − ((𝑢‘3) · (𝑣‘2))), if(𝑘 = 2, (((𝑢‘3) · (𝑣‘1)) − ((𝑢‘1) · (𝑣‘3))), (((𝑢‘1) · (𝑣‘2)) − ((𝑢‘2) · (𝑣‘1)))))))
Syntax	ctripp 50661	Extend class notation to include the scalar triple product of 3-dimensional real coordinate vectors. (Contributed by Jiamin Zhao, 31-Jul-2026.)
class tripp
Definition	df-tripp 50662*	Define the scalar triple product of three 3-dimensional real coordinate vectors as the dot product of the first vector with the cross product of the other two. Vectors are represented as functions on (1...3). Apply as (𝑦(tripp‘𝑥)𝑧). (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ tripp = (𝑥 ∈ (ℝ ↑m (1...3)) ↦ (𝑦 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝑥‘𝑘) · ((𝑦⊠𝑧)‘𝑘))))))
Theorem	crosspval 50663*	Value of the cross product of two 3-dimensional real coordinate vectors as a function on (1...3). (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ ((𝐴 ∈ (ℝ ↑m (1...3)) ∧ 𝐵 ∈ (ℝ ↑m (1...3))) → (𝐴⊠𝐵) = (𝑘 ∈ (1...3) ↦ if(𝑘 = 1, (((𝐴‘2) · (𝐵‘3)) − ((𝐴‘3) · (𝐵‘2))), if(𝑘 = 2, (((𝐴‘3) · (𝐵‘1)) − ((𝐴‘1) · (𝐵‘3))), (((𝐴‘1) · (𝐵‘2)) − ((𝐴‘2) · (𝐵‘1)))))))
Theorem	crosspcle1i 50664	Closure of the first component of the cross product's Sarrus expansion. (A helper for crosspclifi 50667 and crosspv1i 50669.) (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (((𝐴‘2) · (𝐵‘3)) − ((𝐴‘3) · (𝐵‘2))) ∈ ℝ
Theorem	crosspcle2i 50665	Closure of the second component of the cross product's Sarrus expansion. (A helper for crosspclifi 50667 and crosspv2i 50670.) (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (((𝐴‘3) · (𝐵‘1)) − ((𝐴‘1) · (𝐵‘3))) ∈ ℝ
Theorem	crosspcle3i 50666	Closure of the third component of the cross product's Sarrus expansion. (A helper for crosspclifi 50667 and crosspv3i 50671.) (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (((𝐴‘1) · (𝐵‘2)) − ((𝐴‘2) · (𝐵‘1))) ∈ ℝ
Theorem	crosspclifi 50667*	Closure of the three-way Sarrus case split used in the cross product's mapping rule. (A helper for crosspcli 50668.) (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ if(𝑘 = 1, (((𝐴‘2) · (𝐵‘3)) − ((𝐴‘3) · (𝐵‘2))), if(𝑘 = 2, (((𝐴‘3) · (𝐵‘1)) − ((𝐴‘1) · (𝐵‘3))), (((𝐴‘1) · (𝐵‘2)) − ((𝐴‘2) · (𝐵‘1))))) ∈ ℝ
Theorem	crosspcli 50668	Closure of the cross product: the cross product of two 3-dimensional real coordinate vectors is again such a vector. (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (𝐴⊠𝐵) ∈ (ℝ ↑m (1...3))
Theorem	crosspv1i 50669	Value of the first component of the cross product. (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ ((𝐴⊠𝐵)‘1) = (((𝐴‘2) · (𝐵‘3)) − ((𝐴‘3) · (𝐵‘2)))
Theorem	crosspv2i 50670	Value of the second component of the cross product. (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ ((𝐴⊠𝐵)‘2) = (((𝐴‘3) · (𝐵‘1)) − ((𝐴‘1) · (𝐵‘3)))
Theorem	crosspv3i 50671	Value of the third component of the cross product. (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ ((𝐴⊠𝐵)‘3) = (((𝐴‘1) · (𝐵‘2)) − ((𝐴‘2) · (𝐵‘1)))
Theorem	crosspdot0i 50672*	Unfold the curried scalar triple product application into an explicit group sum. (A helper for crosspdoti 50674.) (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐶 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (𝐵(tripp‘𝐴)𝐶) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))))
Theorem	crosspdotsumi 50673*	Expand the group sum over (1...3) into an explicit three-term sum. (A helper for crosspdoti 50674.) (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐶 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) = (((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + (((𝐴‘2) · ((𝐵⊠𝐶)‘2)) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3))))
Theorem	crosspdoti 50674	Value of the scalar triple product, expanded into the standard six-term Sarrus polynomial. (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐶 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (𝐵(tripp‘𝐴)𝐶) = (((((𝐴‘1) · (𝐵‘2)) · (𝐶‘3)) − (((𝐴‘1) · (𝐵‘3)) · (𝐶‘2))) + (((((𝐴‘2) · (𝐵‘3)) · (𝐶‘1)) − (((𝐴‘2) · (𝐵‘1)) · (𝐶‘3))) + ((((𝐴‘3) · (𝐵‘1)) · (𝐶‘2)) − (((𝐴‘3) · (𝐵‘2)) · (𝐶‘1)))))
Theorem	crosspalti 50675*	Antisymmetry of the cross product: swapping the two vectors negates the result. (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 𝐴 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝐵 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (𝐴⊠𝐵) = (𝑘 ∈ (1...3) ↦ -((𝐵⊠𝐴)‘𝑘))
Theorem	crossp3i 50676*	The vector triple product expansion (BAC-CAB rule): the cross product of 𝑋 with (𝑌⊠𝑍) equals 𝑌 scaled by the dot product of 𝑋 and 𝑍, minus 𝑍 scaled by the dot product of 𝑋 and 𝑌. The dot products are written out as explicit three-term sums of component products, matching the pointwise style of df-crossp 50660 rather than introducing a separate dot product operator. (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 𝑋 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝑌 ∈ (ℝ ↑m (1...3))    &   ⊢ 𝑍 ∈ (ℝ ↑m (1...3))    ⇒   ⊢ (𝑋⊠(𝑌⊠𝑍)) = (𝑘 ∈ (1...3) ↦ ((((((𝑋‘1) · (𝑍‘1)) + ((𝑋‘2) · (𝑍‘2))) + ((𝑋‘3) · (𝑍‘3))) · (𝑌‘𝑘)) − (((((𝑋‘1) · (𝑌‘1)) + ((𝑋‘2) · (𝑌‘2))) + ((𝑋‘3) · (𝑌‘3))) · (𝑍‘𝑘))))
21.54  Mathbox for Kunhao Zheng
21.54.1  Weighted AM-GM inequality
Theorem	amgmwlem 50677	Weighted version of amgmlem 27163. (Contributed by Kunhao Zheng, 19-Jun-2021.)
⊢ 𝑀 = (mulGrp‘ℂfld)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ+)    &   ⊢ (𝜑 → 𝑊:𝐴⟶ℝ+)    &   ⊢ (𝜑 → (ℂfld Σg 𝑊) = 1)    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝐹 ∘f ↑𝑐𝑊)) ≤ (ℂfld Σg (𝐹 ∘f · 𝑊)))
Theorem	amgmlemALT 50678	Alternate proof of amgmlem 27163 using amgmwlem 50677. (Contributed by Kunhao Zheng, 20-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑀 = (mulGrp‘ℂfld)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ+)    ⇒   ⊢ (𝜑 → ((𝑀 Σg 𝐹)↑𝑐(1 / (♯‘𝐴))) ≤ ((ℂfld Σg 𝐹) / (♯‘𝐴)))
Theorem	amgmw2d 50679	Weighted arithmetic-geometric mean inequality for 𝑛 = 2 (compare amgm2d 44952). (Contributed by Kunhao Zheng, 20-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑃 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑄 ∈ ℝ+)    &   ⊢ (𝜑 → (𝑃 + 𝑄) = 1)    ⇒   ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
Theorem	young2d 50680	Young's inequality for 𝑛 = 2, a direct application of amgmw2d 50679. (Contributed by Kunhao Zheng, 6-Jul-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑃 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑄 ∈ ℝ+)    &   ⊢ (𝜑 → ((1 / 𝑃) + (1 / 𝑄)) = 1)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ≤ (((𝐴↑𝑐𝑃) / 𝑃) + ((𝐵↑𝑐𝑄) / 𝑄)))

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