P. 510 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50901-51000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dvcot 50901	Derivative of the cotangent function. (Contributed by Jon Pennant, 28-Aug-2026.)
⊢ (ℂ D cot) = (𝑥 ∈ dom cot ↦ -((csc‘𝑥)↑2))
21.55.5  Identities for "if" Utility theorems for "if".
Theorem	ifnmfalse 50902	If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse 4490 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 ∉ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷)
21.55.6  Logarithms generalized to arbitrary base using ` logb ` Most of this subsection was moved to main set.mm, section "Logarithms to an arbitrary base".
Theorem	logb2aval 50903	Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the 2-argument form logb ⟨𝐵, 𝑋⟩ (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ( logb ‘⟨𝐵, 𝑋⟩) = ((log‘𝑋) / (log‘𝐵)))
21.55.7  Logarithm laws generalized to an arbitrary base - log_ Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear. This supports the notational form ((log_‘𝐵)‘𝑋); that looks a little more like traditional notation, but is different from other 2-parameter functions. E.g., ((log_‘;10)‘;;100) = 2. This form is less convenient to work with inside set.mm as compared to the (𝐵 logb 𝑋) form defined separately.
Syntax	clog- 50904	Extend class notation to include the logarithm generalized to an arbitrary base.
class log_
Definition	df-logbALT 50905*	Define the log_ operator. This is the logarithm generalized to an arbitrary base. It can be used as ((log_‘𝐵)‘𝑋) for "log base B of X". This formulation suggested by Mario Carneiro. (Contributed by David A. Wheeler, 14-Jul-2017.) (New usage is discouraged.)
⊢ log_ = (𝑏 ∈ (ℂ ∖ {0, 1}) ↦ (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝑏))))
21.55.8  Formally define notions such as reflexivity EXPERIMENTAL. Several terms are used in comments but not directly defined in set.mm. For example, there are proofs that a number of specific relations are reflexive, but there is no formal definition of what being reflexive actually *means*. Stating the relationships directly, instead of defining a broader property such as being reflexive, can reduce proof size (because the definition of that property does not need to be expanded later). A disadvantage, however, is that there are several terms that are widely used in comments but do not have a clear formal definition. Here we define wffs that formally define some of these key terms. The intent isn't to use these directly, but to instead provide a clear formal definition of widely-used mathematical terminology (we even use this terminology within the comments of set.mm itself). We could define these using extensible structures, but doing so appears overly restrictive. These definitions don't require the use of extensible structures; requiring something to be in an extensible structure to use them is too restrictive. Even if an extensible structure is already in use, it may in use for other things. For example, in geometry, there is a "less-than" relation, but while the geometry itself is an extensible structure, we would have to build a new structure to state "the geometric less-than relation is transitive" (which is more work than it's probably worth). By creating definitions that aren't tied to extensible structures we create definitions that can be applied to anything, including extensible structures, in whatever way we'd like. BJ suggests that it might be better to define these as functions. There are many advantages to doing that, but they won't work for proper classes. I'm currently trying to also support proper classes, so I have not taken that approach, but if that turns out to be unreasonable then BJ's approach is very much worth considering. Examples would be: BinRel = (𝑥 ∈ V ↦ {𝑟 ∣ 𝑟 ⊆ (𝑥 × 𝑥)}), ReflBinRel = (𝑥 ∈ V ↦ {𝑟 ∈ ( BinRel ‘𝑥) ∣ ( I ↾ 𝑥) ⊆ 𝑟}), and IrreflBinRel = (𝑥 ∈ V ↦ {𝑟 ∈ ( BinRel ‘𝑥) ∣ (𝑟 ∩ ( I ↾ 𝑥)) = ∅}). For more discussion see: https://github.com/metamath/set.mm/pull/1286
Syntax	wreflexive 50906	Extend wff definition to include "Reflexive" applied to a class, which is true iff class R is a reflexive relation over the set A. See df-reflexive 50907. (Contributed by David A. Wheeler, 1-Dec-2019.)
wff 𝑅 Reflexive 𝐴
Definition	df-reflexive 50907*	Define reflexive relation; relation 𝑅 is reflexive over the set 𝐴 iff ∀𝑥 ∈ 𝐴𝑥𝑅𝑥. (Contributed by David A. Wheeler, 1-Dec-2019.)
⊢ (𝑅 Reflexive 𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥))
Syntax	wirreflexive 50908	Extend wff definition to include "Irreflexive" applied to a class, which is true iff class R is an irreflexive relation over the set A. See df-irreflexive 50909. (Contributed by David A. Wheeler, 1-Dec-2019.)
wff 𝑅 Irreflexive 𝐴
Definition	df-irreflexive 50909*	Define irreflexive relation; relation 𝑅 is irreflexive over the set 𝐴 iff ∀𝑥 ∈ 𝐴¬ 𝑥𝑅𝑥. Note that a relation can be neither reflexive nor irreflexive. (Contributed by David A. Wheeler, 1-Dec-2019.)
⊢ (𝑅 Irreflexive 𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
21.55.9  Algebra helpers This is an experimental approach to make it clearer (and easier) to do basic algebra in set.mm. These little theorems support basic algebra on equations at a slightly higher conceptual level. Instead of always having to "build up" equivalent expressions for one side of an equation, these theorems allow you to directly manipulate an equality. These higher-level steps lead to easier to understand proofs when they can be used, as well as proofs that are slightly shorter (when measured in steps). There are disadvantages. In particular, this approach requires many theorems (for many permutations to provide all of the operations). It can also only handle certain cases; more complex approaches must still be approached by "building up" equalities as is done today. However, I expect that we can create enough theorems to make it worth doing. I'm trying this out to see if this is helpful and if the number of permutations is manageable. To commute LHS for addition, use addcomli 11502. We might want to switch to a naming convention like addcomli 11502.
Theorem	mvlraddi 50910	Move the right term in a sum on the LHS to the RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 + 𝐵) = 𝐶    ⇒   ⊢ 𝐴 = (𝐶 − 𝐵)
Theorem	assraddsubi 50911	Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐷 ∈ ℂ    &   ⊢ 𝐴 = ((𝐵 + 𝐶) − 𝐷)    ⇒   ⊢ 𝐴 = (𝐵 + (𝐶 − 𝐷))
Theorem	joinlmuladdmuli 50912	Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷    ⇒   ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷
Theorem	joinlmulsubmuld 50913	Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → ((𝐴 · 𝐵) − (𝐶 · 𝐵)) = 𝐷)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐶) · 𝐵) = 𝐷)
Theorem	joinlmulsubmuli 50914	Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ ((𝐴 · 𝐵) − (𝐶 · 𝐵)) = 𝐷    ⇒   ⊢ ((𝐴 − 𝐶) · 𝐵) = 𝐷
Theorem	mvlrmuld 50915	Move the right term in a product on the LHS to the RHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    &   ⊢ (𝜑 → (𝐴 · 𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = (𝐶 / 𝐵))
Theorem	mvlrmuli 50916	Move the right term in a product on the LHS to the RHS, inference form. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    &   ⊢ (𝐴 · 𝐵) = 𝐶    ⇒   ⊢ 𝐴 = (𝐶 / 𝐵)
21.55.10  Algebra helper examples Examples using the algebra helpers.
Theorem	i2linesi 50917	Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use inference form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐷 ∈ ℂ    &   ⊢ 𝑋 ∈ ℂ    &   ⊢ 𝑌 = ((𝐴 · 𝑋) + 𝐵)    &   ⊢ 𝑌 = ((𝐶 · 𝑋) + 𝐷)    &   ⊢ (𝐴 − 𝐶) ≠ 0    ⇒   ⊢ 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶))
Theorem	i2linesd 50918	Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use deduction form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 15-Oct-2018.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑌 = ((𝐴 · 𝑋) + 𝐵))    &   ⊢ (𝜑 → 𝑌 = ((𝐶 · 𝑋) + 𝐷))    &   ⊢ (𝜑 → (𝐴 − 𝐶) ≠ 0)    ⇒   ⊢ (𝜑 → 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶)))
21.55.11  Formal methods "surprises" Prove that some formal expressions using classical logic have meanings that might not be obvious to some lay readers. I find these are common mistakes and are worth pointing out to new people. In particular we prove alimp-surprise 50919, empty-surprise 50921, and eximp-surprise 50923.
Theorem	alimp-surprise 50919	Demonstrate that when using "for all" and material implication the consequent can be both always true and always false if there is no case where the antecedent is true. Those inexperienced with formal notations of classical logic can be surprised with what "for all" and material implication do together when the implication's antecedent is never true. This can happen, for example, when the antecedent is set membership but the set is the empty set (e.g., 𝑥 ∈ 𝑀 and 𝑀 = ∅). This is perhaps best explained using an example. The sentence "All Martians are green" would typically be represented formally using the expression ∀𝑥(𝜑 → 𝜓). In this expression 𝜑 is true iff 𝑥 is a Martian and 𝜓 is true iff 𝑥 is green. Similarly, "All Martians are not green" would typically be represented as ∀𝑥(𝜑 → ¬ 𝜓). However, if there are no Martians (¬ ∃𝑥𝜑), then both of those expressions are true. That is surprising to the inexperienced, because the two expressions seem to be the opposite of each other. The reason this occurs is because in classical logic the implication (𝜑 → 𝜓) is equivalent to ¬ 𝜑 ∨ 𝜓 (as proven in imor 867). When 𝜑 is always false, ¬ 𝜑 is always true, and an or with true is always true. Here are a few technical notes. In this notation, 𝜑 and 𝜓 are predicates that return a true or false value and may depend on 𝑥. We only say may because it actually doesn't matter for our proof. In Metamath this simply means that we do not require that 𝜑, 𝜓, and 𝑥 be distinct (so 𝑥 can be part of 𝜑 or 𝜓). In natural language the term "implies" often presumes that the antecedent can occur in at one least circumstance and that there is some sort of causality. However, exactly what causality means is complex and situation-dependent. Modern logic typically uses material implication instead; this has a rigorous definition, but it is important for new users of formal notation to precisely understand it. There are ways to solve this, e.g., expressly stating that the antecedent exists (see alimp-no-surprise 50920) or using the allsome quantifier (df-als 50927) . For other "surprises" for new users of classical logic, see empty-surprise 50921 and eximp-surprise 50923. (Contributed by David A. Wheeler, 17-Oct-2018.)
⊢ ¬ ∃𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓))
Theorem	alimp-no-surprise 50920	There is no "surprise" in a for-all with implication if there exists a value where the antecedent is true. This is one way to prevent for-all with implication from allowing anything. For a contrast, see alimp-surprise 50919. The allsome quantifier also counters this problem, see df-als 50927. (Contributed by David A. Wheeler, 27-Oct-2018.)
⊢ ¬ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑)
Theorem	empty-surprise 50921	Demonstrate that when using restricted "for all" over a class the expression can be both always true and always false if the class is empty. Those inexperienced with formal notations of classical logic can be surprised with what restricted "for all" does over an empty set. It is important to note that ∀𝑥 ∈ 𝐴𝜑 is simply an abbreviation for ∀𝑥(𝑥 ∈ 𝐴 → 𝜑) (per df-ral 3077). Thus, if 𝐴 is the empty set, this expression is always true regardless of the value of 𝜑 (see alimp-surprise 50919). If you want the expression ∀𝑥 ∈ 𝐴𝜑 to not be vacuously true, you need to ensure that set 𝐴 is inhabited (e.g., ∃𝑥 ∈ 𝐴). (Technical note: You can also assert that 𝐴 ≠ ∅; this is an equivalent claim in classical logic as proven in n0 4299, but in intuitionistic logic the statement 𝐴 ≠ ∅ is a weaker claim than ∃𝑥 ∈ 𝐴.) Some materials on logic (particularly those that discuss "syllogisms") are based on the much older work by Aristotle, but Aristotle expressly excluded empty sets from his system. Aristotle had a specific goal; he was trying to develop a "companion-logic" for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature... This is why he leaves no room for such nonexistent entities in his logic." (Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy, http://www.iep.utm.edu/aris-log/ 4299). While this made sense for his purposes, it is less flexible than modern (classical) logic which does permit empty sets. If you wish to make claims that require a nonempty set, you must expressly include that requirement, e.g., by stating ∃𝑥𝜑. Examples of proofs that do this include barbari 2693, celaront 2695, and cesaro 2702. For another "surprise" for new users of classical logic, see alimp-surprise 50919 and eximp-surprise 50923. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	empty-surprise2 50922	"Prove" that false is true when using a restricted "for all" over the empty set, to demonstrate that the expression is always true if the value ranges over the empty set. Those inexperienced with formal notations of classical logic can be surprised with what restricted "for all" does over an empty set. We proved the general case in empty-surprise 50921. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1597); the problem is that restricted "for all" works in ways that might seem counterintuitive to the inexperienced when given an empty set. Solutions to this can include requiring that the set not be empty or by using the allsome quantifier df-rals 50928. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ⊥
Theorem	eximp-surprise 50923	Show what implication inside "there exists" really expands to (using implication directly inside "there exists" is usually a mistake). Those inexperienced with formal notations of classical logic may use expressions combining "there exists" with implication. That is usually a mistake, because as proven using imor 867, such an expression can be rewritten using not with or - and that is often not what the author intended. New users of formal notation who use "there exists" with an implication should consider if they meant "and" instead of "implies". A stark example is shown in eximp-surprise2 50924. See also alimp-surprise 50919 and empty-surprise 50921. (Contributed by David A. Wheeler, 17-Oct-2018.)
⊢ (∃𝑥(𝜑 → 𝜓) ↔ ∃𝑥(¬ 𝜑 ∨ 𝜓))
Theorem	eximp-surprise2 50924	Show that "there exists" with an implication is always true if there exists a situation where the antecedent is false. Those inexperienced with formal notations of classical logic may use expressions combining "there exists" with implication. This is usually a mistake, because that combination does not mean what an inexperienced person might think it means. For example, if there is some object that does not meet the precondition 𝜑, then the expression ∃𝑥(𝜑 → 𝜓) as a whole is always true, no matter what 𝜓 is (𝜓 could even be false, ⊥). New users of formal notation who use "there exists" with an implication should consider if they meant "and" instead of "implies". See eximp-surprise 50923, which shows what implication really expands to. See also empty-surprise 50921. (Contributed by David A. Wheeler, 18-Oct-2018.)
⊢ ∃𝑥 ¬ 𝜑    ⇒   ⊢ ∃𝑥(𝜑 → 𝜓)
21.55.12  Allsome quantifier These are definitions and proofs involving the "allsome" quantifier (aka "all some"). In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like ∀𝑥𝜑 → 𝜓 do not imply that 𝜑 is ever true, leading to vacuous truths. See alimp-surprise 50919 and empty-surprise 50921 as examples of the problem. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem. The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines. I do not expect this to be used much in Metamath, because in Metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it. The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow ∀∃𝑥(𝜑 → 𝜓), and when restricted (applied to a class) we allow ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓). The first symbol after the setvar variable must always be ∈ if it is the form applied to a class, and since ∈ cannot begin a wff, it is unambiguous. The → looks like it would be a problem because 𝜑 or 𝜓 might include implications, but any implication arrow → within any wff must be surrounded by parentheses, so only the implication arrow of ∀∃ can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax. Naming: "als" is allsome. The form restricted to a class is prefixed with "r", following the way set.mm names the restricted quantifiers it is built from: ∀ gives df-ral 3077 and ∃ gives df-rex 3087, so df-als 50927 (the general form) gives df-rals 50928 (the restricted form). Earlier versions of this material differed, so old references may not match. They wrote the quantifier as an "inverted A" followed by an exclamation point, and they named the general form df-alsi and the restricted form df-alsc. The symbol is now an "inverted A" followed by a "backwards E", which more readers can correctly guess without being taught it. The restricted definition also changed, and the older one was a mistake; see df-rals 50928 for what was wrong with it. Soundness of the two definitions below. Definitions are required to be eliminable and conservative (see the section comment for df-bi 210). Both df-als 50927 and df-rals 50928 meet these requirements directly, and so neither needs a justification theorem. Each is stated as a biconditional whose left side is a new syntax construct (wals 50925 or wrals 50926) applied to distinct metavariables, and whose right side uses only constructs introduced earlier (∀, ∃, ∧, →, and the restricted quantifiers df-ral 3077 and df-rex 3087). Any occurrence of the new construct can therefore 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. Note in particular that every variable occurring on the right side already occurs on the left side, so no dummy variable is introduced. A justification theorem is needed only when that fails, that is, when a definition introduces a dummy variable and the choice of that variable must be shown not to matter (as in eujust 2596 for df-eu 2594), or when the definition cannot use ↔ because it is defining ↔ itself (as in bijust 208 for df-bi 210). Neither case arises here. The restricted quantifier definitions df-ral 3077 and df-rex 3087 have the same shape as these and likewise need no justification. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html 3087 I hope that others will eventually agree that allsome is awesome.
Syntax	wals 50925	Extend wff definition to include "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and there is at least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff ∀∃𝑥(𝜑 → 𝜓)
Syntax	wrals 50926	Extend wff definition to include "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. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓)
Definition	df-als 50927	Define "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true and there is at least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
Definition	df-rals 50928	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 50929	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 50930	Introduction rule: "all some" holds if the "for all" part holds and the antecedent has a witness. This is the converse of als1d 50932 and als2d 50933 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 50931	Introduction rule for "all some" restricted to a class. This is the converse of rals1d 50934 and rals2d 50935 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 50932	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 50933	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 50934	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 50935	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 50936*	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 50937	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 50945, 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 50919 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 50938	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 50937. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	alsbii 50939	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 50940	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 50941	Deduction form of alsbii 50939. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (∀∃𝑥(𝜓 → 𝜒) ↔ ∀∃𝑥(𝜃 → 𝜏)))
Theorem	nfals 50942	Bound-variable hypothesis builder for "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)
Theorem	nfrals 50943*	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 50944*	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 50945	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 50927; the proof itself builds on alimp-no-surprise 50920. For a contrast, see alimp-surprise 50919. (Contributed by David A. Wheeler, 27-Oct-2018.)
⊢ ¬ (∀∃𝑥(𝜑 → 𝜓) ∧ ∀∃𝑥(𝜑 → ¬ 𝜓))
Theorem	rals-no-surprise 50946	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 50945, and follows from it by dfrals2 50929. 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 50922; that is the point of allsome. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ ¬ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓))
Theorem	ralrals 50947	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 50953. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	rexrals 50948	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 50954. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)))
Theorem	alsanmo 50949	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 50950	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 50949. (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 50951*	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 50952*	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 50953*	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 50947 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 50954*	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 50948 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 50955*	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 50956*	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 50957*	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 ≠ ∅))
21.55.13  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 2594) 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 50919 and empty-surprise 50921. 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 50975 for the exact relationship between the two and alseueu 50976 for the one direction that does hold. Naming: "alseu" is allsome ("als", as in df-als 50927) extended with "exactly one" ("eu", as in df-eu 2594). The form restricted to a class is prefixed with "r", following df-rals 50928 and df-reu 3366, giving df-ralseu 50961. Soundness: df-alseu 50960 and df-ralseu 50961 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 50958 or wralseu 50959) applied to distinct metavariables, and whose right side uses only constructs introduced earlier (∀, ∃!, ∧, →, and the restricted quantifiers df-ral 3077 and df-reu 3366), 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 3366
Syntax	walseu 50958	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 50959	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 50960	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 50961	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 50962	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 50929. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
Theorem	alseuals 50963	"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 50977 is proved. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (∀∃!𝑥(𝜑 → 𝜓) → ∀∃𝑥(𝜑 → 𝜓))
Theorem	ralseurals 50964	"All some one" restricted to a class implies "all some" restricted to that class. Restricted counterpart of alseuals 50963. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓))
Theorem	alseud 50965	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 50967 and alseu2d 50968 taken together. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))    &   ⊢ (𝜑 → ∃!𝑥𝜓)    ⇒   ⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))
Theorem	ralseud 50966	Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d 50969 and ralseu2d 50970 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 50967	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 50968	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 50969	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 50970	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 50971	Congruence: equivalents may be substituted inside an "all some one". This is the "all some one" counterpart of alsbii 50939. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ ∀∃!𝑥(𝜒 → 𝜃))
Theorem	ralseubii 50972	Congruence for "all some one" restricted to a class. This is the "all some one" counterpart of ralsbii 50940. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥 ∈ 𝐴(𝜒 → 𝜃))
Theorem	nfalseu 50973*	Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 50942. Unlike nfals 50942 it requires 𝑥 and 𝑦 to be disjoint, because the corresponding builder for ∃! is nfeuw 2618, which requires it; the version without that requirement, nfeu 2619, depends on ax-13 2401 and its use is discouraged. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃!𝑦(𝜑 → 𝜓)
Theorem	nfralseu 50974*	Bound-variable hypothesis builder for "all some one" restricted to a class. This is the "all some one" counterpart of nfrals 50943. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃!𝑦 ∈ 𝐴(𝜑 → 𝜓)
Theorem	dfalseu2 50975	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 50960 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 50976 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 50976	"The 𝜑 is 𝜓 " implies that exactly one thing is both 𝜑 and 𝜓. This is the half of dfalseu2 50975 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 50977	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 50945 by alseuals 50963. For a contrast, see alimp-surprise 50919. (Contributed by David A. Wheeler, 21-Jul-2026.)
⊢ ¬ (∀∃!𝑥(𝜑 → 𝜓) ∧ ∀∃!𝑥(𝜑 → ¬ 𝜓))
21.55.14  Miscellaneous Miscellaneous proofs.
Theorem	5m4e1 50978	Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
⊢ (5 − 4) = 1
Theorem	2p2ne5 50979	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 50980	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 50981	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)
21.55.15  Theorems about algebraic numbers
Theorem	aacllem 50982*	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)
21.56  Mathbox for Mingli Yuan
Theorem	wrdf1d 50983	A one-to-one word maps its domain into its alphabet. (Contributed by Mingli Yuan, 11-Aug-2026.)
⊢ (𝜑 → 𝑊 ∈ Word 𝐷)    &   ⊢ (𝜑 → Fun ◡𝑊)    ⇒   ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
21.57  Mathbox for Jiamin Zhao
21.57.1  Cross product and scalar triple product in RR^3
Theorem	1ne3 50984	1 is not equal to 3. (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 1 ≠ 3
Theorem	2ne3 50985	2 is not equal to 3. (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 2 ≠ 3
Theorem	1elfz13 50986	Membership of 1 in the integer interval ( 1 ... 3 ). (Suggested by avekens.) (Contributed by Jiamin Zhao, 1-Aug-2026.) (Proof shortened by Jiamin Zhao, 13-Aug-2026.)
⊢ 1 ∈ (1...3)
Theorem	2elfz13 50987	Membership of 2 in the integer interval ( 1 ... 3 ). (Suggested by tirix.) (Contributed by Jiamin Zhao, 1-Aug-2026.) (Proof shortened by Jiamin Zhao, 13-Aug-2026.)
⊢ 2 ∈ (1...3)
Theorem	3elfz13 50988	Membership of 3 in the integer interval ( 1 ... 3 ). (Contributed by Jiamin Zhao, 1-Aug-2026.)
⊢ 3 ∈ (1...3)
Theorem	rr3fvcl 50989	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	rr3fv1cld 50990	First component of a 3-dimensional real coordinate vector is real. (Contributed by Jiamin Zhao, 10-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (𝐴‘1) ∈ ℝ)
Theorem	rr3fv2cld 50991	Second component of a 3-dimensional real coordinate vector is real. (Contributed by Jiamin Zhao, 10-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (𝐴‘2) ∈ ℝ)
Theorem	rr3fv3cld 50992	Third component of a 3-dimensional real coordinate vector is real. (Contributed by Jiamin Zhao, 10-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (𝐴‘3) ∈ ℝ)
Syntax	ccrossp 50993	Extend class notation to include the cross product operation. (Contributed by Jiamin Zhao, 31-Jul-2026.)
class ⊠
Definition	df-crossp 50994*	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 50995	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 50996*	Define the scalar triple product of three 3-dimensional real coordinate vectors as the dot product of the first vector 𝑥 with the cross product of the other two (𝑦 and 𝑧). Apply as (𝑦(tripp‘𝑥)𝑧). Vectors are represented as functions on (1...3). (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 50997*	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	crosspcle1d 50998	Closure of the first component of the cross product's coordinate formula. (Contributed by Jiamin Zhao, 11-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (((𝐴‘2) · (𝐵‘3)) − ((𝐴‘3) · (𝐵‘2))) ∈ ℝ)
Theorem	crosspcle2d 50999	Closure of the second component of the cross product's coordinate formula. (Contributed by Jiamin Zhao, 11-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (((𝐴‘3) · (𝐵‘1)) − ((𝐴‘1) · (𝐵‘3))) ∈ ℝ)
Theorem	crosspcle3d 51000	Closure of the third component of the cross product's coordinate formula. (Contributed by Jiamin Zhao, 11-Aug-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   ⊢ (𝜑 → (((𝐴‘1) · (𝐵‘2)) − ((𝐴‘2) · (𝐵‘1))) ∈ ℝ)