P. 379 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37801-37900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wl-df2-3mintru2 37801	The adder carry in disjunctive normal form. An alternative highly symmetric definition emphasizing the independence of order of the inputs 𝜑, 𝜓 and 𝜒. Copy of cador 1610. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 12-Jun-2024.)
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))
Theorem	wl-df3-3mintru2 37802	The adder carry in conjunctive normal form. An alternative highly symmetric definition emphasizing the independence of order of the inputs 𝜑, 𝜓 and 𝜒. Copy of cadan 1611. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 18-Jun-2024.)
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))
Theorem	wl-df4-3mintru2 37803	An alternative definition of the adder carry. Copy of df-cad 1609. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 19-Jun-2024.)
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))
Theorem	wl-1mintru1 37804	Using the recursion formula: "(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" ) for "1-mintru-1" (meaning "at least 1 out of 1 input is true") by plugging in n = 0, m = 0, and simplifying. The expressions "0-mintru-0" and "0-mintru-1" are base cases of the recursion, meaning "in a sequence of zero inputs, at least 0 / 1 input is true", respectively equivalent to ⊤ / ⊥. Negating an "n-mintru1" operation means: All n inputs 𝜑.. 𝜃 are false. This is also conveniently expressed as ¬ (𝜑 ∨.. ∨ 𝜃). Applying this idea here (n = 1) yields the obvious result that in an input sequence of size 1 only then all will be false, if its single input is. (Contributed by Wolf Lammen, 10-May-2024.)
⊢ (if-(𝜒, ⊤, ⊥) ↔ 𝜒)
Theorem	wl-1mintru2 37805	Using the recursion formula: "(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" ) for "1-mintru-2" (meaning "at least 2 out of a single input are true") by plugging in n = 0, m = 1, and simplifying. The expressions "0-mintru-1" and "0-mintru-2" are base cases of the recursion, meaning "in a sequence of zero inputs at least 1 / 2 input is true", evaluate both to ⊥. Since no sequence of inputs has a longer subsequence of whatever property, the resulting ⊥ is to be expected. Negating a "n-mintru2" operation has an interesting interpretation: at most one input is true, so all inputs exclude each other mutually. Such an exclusion is expressed by a NAND operation (𝜑 ⊼ 𝜓), not by a XOR. Applying this idea here (n = 1) leads to the obvious "In a single input sequence 'at most one is true' always holds". (Contributed by Wolf Lammen, 10-May-2024.)
⊢ (if-(𝜒, ⊥, ⊥) ↔ ⊥)
Theorem	wl-2mintru1 37806	Using the recursion formula "(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" ) for "2-mintru-1" (meaning "at least 1 out of 2 inputs is true") by plugging in n = 1, m = 0, and simplifying. The expression "1-mintru-0" is a base case (meaning at least zero inputs out of 1 are true), evaluating to ⊤, and wl-1mintru1 37804 shows "1-mintru-1" is equivalent to the only input. Negating an "n-mintru1" operation means: All n inputs 𝜑.. 𝜃 are false. This is also conveniently expressed as ¬ (𝜑 ∨.. ∨ 𝜃), in accordance with the result here. (Contributed by Wolf Lammen, 10-May-2024.)
⊢ (if-(𝜓, ⊤, 𝜒) ↔ (𝜓 ∨ 𝜒))
Theorem	wl-2mintru2 37807	Using the recursion formula "(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" ) for "2-mintru-2" (meaning "2 out of 2 inputs are true") by plugging in n = 1, m = 1, and simplifying. See wl-1mintru1 37804 and wl-1mintru2 37805 to see that "1-mintru-1" / "1-mintru-2" evaluate to 𝜒 / ⊥ respectively. Negating a "n-mintru2" operation means 'at most one input is true', so all inputs exclude each other mutually. Such an exclusion is expressed by a NAND operation (𝜑 ⊼ 𝜓), not by a XOR. Applying this idea here (n = 2) yields the expected NAND in case of a pair of inputs. (Contributed by Wolf Lammen, 10-May-2024.)
⊢ (if-(𝜓, 𝜒, ⊥) ↔ (𝜓 ∧ 𝜒))
Theorem	wl-df3maxtru1 37808	Assuming "(n+1)-maxtru1" ↔ ¬ "(n+1)-mintru-2", we can deduce from the recursion formula given in wl-df-3mintru2 37800, that a similiar one "(n+1)-maxtru1" ↔ if-(𝜑,-. "n-mintru-1" , "n-maxtru1" ) is valid for expressing 'at most one input is true'. This can also be rephrased as a mutual exclusivity of propositional expressions (no two of a sequence of inputs can simultaneously be true). Of course, this suggests that all inputs depend on variables 𝜂, 𝜁... Whatever wellformed expression we plugin for these variables, it will render at most one of the inputs true. The here introduced mutual exclusivity is possibly useful for case studies, where we want the cases be sort of 'disjoint'. One can further imagine that a complete case scenario demands that the 'at most' is sharpened to 'exactly one'. This does not impose any difficulty here, as one of the inputs will then be the negation of all others be or'ed. As one input is determined, 'at most one' is sufficient to describe the general form here. Since cadd is an alias for 'at least 2 out of three are true', its negation is under focus here. (Contributed by Wolf Lammen, 23-Jun-2024.)
⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ⊽ 𝜒), (𝜓 ⊼ 𝜒)))
21.22.5  An alternative axiom ~ ax-13
Axiom	ax-wl-13v 37809*	A version of ax13v 2377 with a distinctor instead of a distinct variable condition. Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1912. So essentially this theorem states, that a distinct variable condition between set variables can be replaced with a distinctor expression. (Contributed by Wolf Lammen, 23-Jul-2021.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	wl-ax13lem1 37810*	A version of ax-wl-13v 37809 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
21.22.6  Bootstrapping set theory with classes
Theorem	wl-cleq-0 37811*	Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This and the following texts should be read as explorations rather than as definite statements, open to doubt, alternatives, and reinterpretation. Preface Three specific theorems are under focus in the following pages: df-cleq 2728, df-clel 2811, and df-clab 2715. Only technical concepts necessary to explain these will be introduced, along with a selection of supporting theorems. The three theorems are central to a bootstrapping process that introduces objects into set.mm. We will first examine how Metamath in general creates basic new ideas from scratch, and then look at how these methods are applied specifically to classes, capable of representing objects in set theory. In Zermelo-Fraenkel set theory with the axiom of choice (ZFC), these three theorems are (more or less) independent of each other, which means they can be introduced in different orders. From my own experience, another order has pedagogical advantages: it helps grasping not only the overall concept better, but also the intricate details that I first found difficult to comprehend. Reordering theorems, though syntactically possible, sometimes may cause doubts when intermediate results are not strictly tied to ZFC only. The purpose of set.mm is to provide a formal framework capable of proving the results of ZFC, provided that formulas are properly interpreted. In fact, there is freedom of interpretation. The database set.mm develops from the very beginning, where nothing is assumed or fixed, and gradually builts up to the full abstraction of ZFC. Along the way, results are only preliminary, and one may at any point branch off and pursue a different path toward another variant of set theory. This openess is already visible in axiom ax-mp 5, where the symbol → can be understood as as implication, bi-conditional, or conjunction. The notation and symbol shapes are suggestive, but their interpretation is not mandatory. The point here is that Metamath, as a purely syntactic system, can sometimes allow freedoms, unavailable to semantically fixed systems, which presuppose only a single ultimate goal. (Contributed by Wolf Lammen, 28-Sep-2025.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	wl-cleq-1 37812*	Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This text should be read as an exploration rather than as definite statements, open to doubt, alternatives, and reinterpretation. Grammars and Parsable contents A Metamath program is a text-based tool. Its input consists of primarily human-readable and editable text stored in *.mm files. For automated processing, these files follow a strict structure. This enables automated analysis of their contents, verification of proofs, proof assistance, and the generation of output files - for example, the HTML page of this dummy theorem. A set.mm file contains numerous such structured instructions serving these purposes. This page provides a brief explanation of the general concepts behind structured text, as exemplified by set.mm. The study of structured text originated in linguistics, and later computer science formalized it further for text like data and program code. The rules describing such structures are collectively known as grammar. Metamath also introduces a grammar to support automation and establish a high degree of confidence in its correctness. It could have been described using the terminology of earlier scientific disciplines, but instead it uses its own language. When a text exhibits a sufficiently regular structure, its form can be described by a set of syntax rules, or grammar. Such rules consist of terminal symbols (fixed, literal elements) and non-terminal symbols, which can recursively be expanded using the grammar's rewrite rules. A program component that applies a grammar to text is called a parser. The parser decomposes the text into smaller parts, called syntactic units, while often maintaining contextual information. These units may then be handed over to subsequent components for further processing, such as populating a database or generating output. In these pages, we restrict our attention to strictly formatted material consisting of formulas with logical and mathematical content. These syntactic units are embedded in higher-level structures such as chapters, together with commands that, for example, control the HTML output. Conceptually, the parsing process can be viewed as consisting of two stages. The top-level stage applies a simple built-in grammar to identify its structural units. Each unit is a text region marked on both sides with predefined tokens (in Metamath: keywords), beginning with the escape character "$". Text regions containing logical or mathematical formulas are then passed to a second-stage parser, which applies a different grammar. Unlike the first, this grammar is not built-in but is dynamically constructed. In what follows, we will ignore the first stage of parsing, since its role is only to extract the relevant material embedded within text primarily intended for human readers. (Contributed by Wolf Lammen, 18-Aug-2025.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	wl-cleq-2 37813*	Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This text should be read as an exploration rather than as definite statements, open to doubt, alternatives, and reinterpretation. Vocabulary Sentence: A sentence or closed form theorem is a theorem without any hypothesis, as opposed to the inference form. Although distinct variable conditions can be viewed as a particular kind of hypothesis, in Metamath they are treated as side conditions, and do not affect formal structure of the theorem. Expressions such as Ⅎ𝑥𝜑 or the distinctor ∀𝑥(𝑥 = 𝑦) play a similar role, but appear as separate hypotheses, or they can simply serve as the first antecedent in a sentence. Bound / Free / Dependent: In formal theories, variables are used to represent objects, allowing for general statements about relations rather than individual cases. In Metamath, for example, mathematical objects are represented by variables of type "setvar". An occurrence of such a variable in any formula 𝜑 is said to be bound. if 𝜑 quantifies that variable, as in ∀𝑥𝜑. Variables not bound by a quantifier are called free. Variables of type "class" are always free, since quantifiers do not apply to them. A free variable often indicates a parameter dependency; however, the formula 𝑥 = 𝑥 shows that this is not necessarily the case. In Metamath, a variable expressing a real dependency is also called "effectively free" (see nfequid 2015, with thanks to SN for pointing out this theorem). Instance: A formula of type "class" that is not a mere variable is called an instance of any "class" type variable. An instance may represent a single object, or it may still describe a family of objects, when variables expressing dependencies occur within that formula. Attribute: Objects possess properties, some of which may uniquely identify them. In logic, properties are also known as attributes. They are described by formulas depending on a variable, which can then be substituted with a specific object. If the resulting statement is true, that particular attribute is said to apply to the object. Multiple objects forming an instance may share common attributes. A variable inherits the attributes of the instance it represents. (Contributed by Wolf Lammen, 12-Oct-2025.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	wl-cleq-3 37814*	Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This text should be read as an exploration rather than as definite statements, open to doubt, alternatives, and reinterpretation. Introducing a New Concept: Well-formed formulas The parser that processes strictly formatted Metamath text with logical or mathematical content constructs its grammar dynamically. New syntax rules are added on the fly whenever they are needed to parse upcoming formulas. Only a minimal set of built-in rules - those required to introduce new grammar rules - is predefined; everything else must be supplied by set.mm itself as syntactic units identified by the first-stage parser. Text regions beginning with the tokens "$c" or "$v", for example, are part of such grammar extensions. We will outline the extension process used when introducing an entirely new concept that cannot build on any prior material. As a simple example, we will trace here the steps involved in defining formulas, the expressions used for hypotheses and statements - a concept first-order logic needs from the very beginning to articulate its ideas. 1. Introduce a type code To mark new formulas and variables later used in theorems, the constant "wff" is reserved. It abbreviates "well-formed formula", a term that already suggests that such formulas must be syntactically valid and therefore parseable to enable automatic processing. 2. Introduce variables Variables with unique names such as 𝜑, 𝜓, ..., intended to later represent formulas of type "wff". In grammar terms, they correspond to non-terminal symbols. These variables are marked with the type "wff". Variables are fundamental in Metamath, since they enable substitution during proofs: variables of a particular type can be consistently replaced by any formula of the same type. In grammar terms, this corresponds to applying a rewrite rule. At this step, however, no rewrite rule exists; we can only substitute one "wff" variable for another. This suffices only for very elementary theorems such as idi 1. In the formal language of Metamath these variables are not interpreted with concrete statements, but serve purely as placeholders for substitution. 3. Add primitive formulas Rewrite rules describing primitive formulas of type "wff" are then added to the grammar. Typically, they describe an operator (a constant in the grammar) applied to one or more variables, possibly of different types (e.g. ∀𝑥𝜑, although at this stage only "wff" is available). Since variables are non-terminal symbols, more complex formulas can be constructed from primitive ones, by consistently replacing variables with any wff formula - whether involving the same operator or different ones introduced by other rewrite rules. Whenever such a replacement introduces variables again, they may in turn recursively be replaced. If an operator takes two variables of type wff, it is called a binary connective in logic. The first such operator encountered is (𝜑 → 𝜓). Based on its token, its intended meaning is material implication, though this interpretation is not fixed from the outset. 4. Specify the properties of primitive formulas Once the biconditional connective is available for formulas, new connectives can be defined by specifying replacement formulas that rely solely on previously introduced material. Such definitions makes it possible to eliminate the definiens. At the very beginning, however, this is not possible for well-formed formulas, since little or no prior material exists. Instead, the semantics of an expression such as (𝜑 → 𝜓) are progressively constrained by axioms - that is, theorems without proof. The first such axiom for material implication is ax-mp 5, with additional axioms following later. (Contributed by Wolf Lammen, 21-Aug-2025.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	wl-cleq-4 37815*	Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This text should be read as an exploration rather than as definite statements, open to doubt, alternatives, and reinterpretation. Introducing a New Concept: Classes In wl-cleq-3 37814 we examined how the basic notion of a well-formed formula is introduced in set.mm. A similar process is used to add the notion of a class to Metamath. This process is somewhat more involved, since two parallel variants are established: sets and the broader notion of classes, which include sets (see 3a. below). In Zermelo-Fraenkel set theory (ZF) classes will serve as a convenient shorthand that simplifies formulas and proofs. Ultimately, only sets - a part of all classes - are intended to exist as actual objects. In the First Order Logic (FOL) portion of set.mm objects themselves are not used - only variables representing them. It is not even assumed that objects must be sets at all. In principle, the universe of discourse could consist of anything - vegetables, text strings, and so on. For this reason, the type code "setvar", used for object variables, is somewhat of a misnomer. Its final meaning - and the name that goes with it - becomes justified only in later developments. We will now revisit the four basic steps presented in wl-cleq-3 37814, this time focusing on object variables and paying special attention to the additional complexities that arise from extending sets to classes. 1. Introduce type codes 1a. Reserve a type code for classes, specifically the grammar constant "class". Initially, this type code applies to class variables and will later, beyond these four steps, also be assigned to formulas that define specific classes, i.e. instances. 1b. Reserve a type code for set variables, represented by the grammar constant "setvar". The name itself indicates that this type code will never be assigned to a formula describing a specific set, but only to variables containing such objects. 2. Introduce variables 2a. Use variables with unique names such as 𝐴, 𝐵, ..., to represent classes. These class variables are assigned the type code "class", ensuring that only formulas of type "class" can be substituted for them during proofs. 2b. Use variables with unique names such as 𝑎, 𝑏 , ..., to represent sets. These variables are assigned the type code "setvar". During a substitution in a proof, a variable of type "setvar" may only be replaced with another variable of this type. No specific formula, or object, of this type exists. 3. Add primitive formulas 3a. Add the rewrite rule cv 1541 to set.mm, allowing variables of type "setvar" to also aquire type "class". In this way, a variable of "setvar" type can serve as a substitute for a class variable. 3b. Add the rewrite rules wcel 2114 and wceq 1542 to set.mm, making the formulas 𝐴 ∈ 𝐵 and 𝐴 = 𝐵 valid well-formed formulas (wff). Since variables of type "setvar" can be substituted for class variables, 𝑥 ∈ 𝑦 and 𝑥 = 𝑦 are also provably valid wff. In the FOL part of set.mm, only these specific formulas play a role and are therefore treated as primitive there. The underlying universe may not contain sets, and the notion of a class is not even be required. Additional mixed-type formulas, such as 𝑥 ∈ 𝐴, 𝐴 ∈ 𝑥, 𝑥 = 𝐴, and 𝐴 = 𝑥 exist. When the theory is later refined to distinguish between sets and classes, the results from FOL remain valid and naturally extend to these mixed cases. These cases will occasionally be examined individually in subsequent discussions. 4. Specify the properties of primitive formulas In FOL in set.mm, the formulas 𝑥 = 𝑦 and 𝑥 ∈ 𝑦 cannot be derived from earlier material, and therefore cannot be defined. Instead, their fundamental properties are established through axioms, namely ax-6 1969 through ax-9 2124, ax-13 2376, and ax-ext 2708. Similarly, axioms establish the properties of the primitive formulas 𝐴 = 𝐵 and 𝐴 ∈ 𝐵, ensuring that they extend the FOL counterparts 𝑥 = 𝑦 and 𝑥 ∈ 𝑦 in a consistent and meaningful way. At the same time, a criterion must be developed to distinguish sets from classes. Since set variables can only be substituted by other set variables, equality must permit the assignment of class terms known to represent sets to those variables. (Contributed by Wolf Lammen, 25-Aug-2025.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	wl-cleq-5 37816*	Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This text should be read as an exploration rather than as definite statements, open to doubt, alternatives, and reinterpretation. Semantics of Equality There is a broadly shared understanding of what equality between objects expresses, extending beyond mathematics or set theory. Equality constitutes an equivalence relation among objects 𝑥, 𝑦, and 𝑧 within the universe under consideration: 1. Reflexivity 𝑥 = 𝑥 2. Symmetry (𝑥 = 𝑦 → 𝑦 = 𝑥) 3. Transitivity ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧) 4. Identity of Indiscernables (Leibniz's Law): distinct (i.e., unequal), objects cannot share all the same properties (or attributes). In formal theories using variables, the attributes of a variable are assumed to mirror those of the instance it denotes. For both variables and objects, items (1) - (4) must either be derived or postulated as axioms. If the theory allows substituting instances for variables, then the equality rules for objects follow directly from those governing variables. However, if variables and instances are formally distinguished, this distinction introduces an additional metatheoretical attribute, relevant for (4). A similar issue arises when equality is considered between different types of variables sharing properties. Such mixed-type equalities are subject to restrictions: reflexivity does not apply, since the two sides represent different kinds of entities. Nevertheless, symmetry and various forms of transitivity typically remain valid, and must be proven or established within the theory. In set.mm formulas express attributes. Therefore, equal instances must behave identically, yielding the same results when substituted into any formula. To verify equality, it suffices to consider only primitive operations involving free variables, since all formulas - once definitions are eliminated - reduce to these. Equality itself introduces no new attribute (an object is always different from all others), and can thus be excluded from this examination. Equality in First Order Logic (FOL) In the FOL component of set.mm, the notion of an "object" is absent. Only set variables are used to formulate theorems, and their attributes - expressed through an unspecified membership operator - are addressed at a later stage. Instead, several axioms address equality directly: ax-6 1969, ax-7 2010, ax-8 2116, ax-9 2124 and ax-12 2185, and ax-13 2376. In practice, restricted versions with distinct variable conditions are used (ax6v 1970, ax12v 2186). The unrestricted forms together with axiom ax-13 2376, allow for the elimination of distinct variable conditions, this benefit is considered too minor for routine use. Equality in FOL is formalized as follows: 2a. Equivalence Relation. Essentially covered by ax-7 2010, with some support of ax-6 1969. 2b. Leibniz's Law for the primitive ∈ operator. Captured by ax-8 2116 and ax-9 2124. 2c. General formulation. Given in sbequ12 2259. 2d. Implicit substitution. Assuming Leibniz's Law holds for a particular expression, various theorems extend its validity to other, derived expressions, often introducing quantification (see for example cbvalvw 2038). The auxiliary axioms ax-10 2147, ax-11 2163, ax-12 are provable (see ax10w 2135, for example) if you can substitute 𝑦 for 𝑥 in a formula 𝜑 that contains no occurrence of 𝑦 and leaves no remaining trace of 𝑥 after substitution. An implicit substitution is then established by setting the resulting formula equivalent to 𝜑 under the assumption 𝑥 = 𝑦. Ordinary FOL substitution [𝑦 / 𝑥]𝜑 is insufficient in this context, since 𝑥 still occurs in the substituted formula. A simple textual replacement of the token 𝑥 by 𝑦 in 𝜑 might seem an intuitive solution, but such operations are out of the formal scope of Metamath. 2e. Axiom of Extensionality. In its elaborated form (axextb 2711), it states that the determining attributes of a set 𝑥 are the elements 𝑧 it contains, as expressed by 𝑧 ∈ 𝑥. This is the only primitive operation relevant for equality between set variables. Equality between classes In set.mm class variables of type "class" are introduced analoguously to set variables. Besides the primitive operations equality and membership, class builders allow other syntactical constructs to substitute for class variables, enabling them to represent class instances. One such builder (cv 1541) allows set variables to replace class variables. Another (df-clab 2715) introduces a class instance, known as class abstraction. Since a class abstraction can freely substitute for a class variable, formulas hold for both alike. Hence, there is no need to distinguish between class variables and abstractions; the term class will denote "class variable or class abstraction". Set variables, however, are treated separately, as they are not of type "class". 3a. Equivalence Relation. Axiom df-cleq 2728, from which class versions of (1a) - (1c) can be derived, guarantees that equality between class variables form an equivalence relation. Since both class abstractions and set variables can substitute for class variables, this equivalence extends to all mixed equalities, including those with set variables, since they automatically convert to classes upon substitution. 3b. Attributes. The primitive operation of membership constitutes the fundamental attributes of a class. Axiom df-clel 2811 reduces possible membership relations between class variables to those between a set variable and a class variable. Axiom df-cleq 2728 extends axextb 2711 to classes, stating that classes are fully determined by their set members. A class builder may introduce a new attribute for classes. An equation involving such a class instance may express this attribute. In the case of the class builder cv 1541, an attribute called sethood is in fact introduced: A class is a set if it can be equated with some set variable. Class abstractions supported by class builder df-clab 2715 also formally introduce attributes. Whether a class can be expressed as an abstraction with a specific predicate may be relevant in analysis. However, since theorem df-clab 2715 is a definition (and hence eliminable), these attributes can also be expressed in other ways. 3c. Conservativity. Because set variables can substitute for class variables, all axioms and definitions must be consistent with theorems in FOL. To ensure this, hypotheses are added to axioms and definitions that mirror the structure of their statement, but with class variables replaced by set variables. Since theorems cannot be applied without first proving their hypotheses, conservativity thus is enforced. 3d. Leibniz's Law. Besides equality membership is (and remains) the only primitive operator between classes. Axioms df-cleq 2728 and df-clel 2811 provide class versions of ax-8 2116 and ax-9 2124, ensuring that membership is consistent with Leibniz's Law. Sethood, being based on mixed-type equality, preserves its value among equal classes. As long as additional class builders beyond those mentioned are only defined, the reasoning given for class abstraction above applies generally, and Leibniz's Law continues to hold. 3e. Backward Compatability. A class 𝐴 equal to a set should be substitutable for a free set variable 𝑥 in any theorem, yielding a valid result, provided 𝑥 and 𝐴 are distinct. Sethood is conveniently expressed by ∃𝑧𝑧 = 𝐴; this assumption is added as an antecedent to the corresponding FOL theorem. However, since direct substitution is disallowed, a deduction version of an FOL theorem cannot be simply converted. Instead, the proof must be replayed, consistently replacing 𝑥 with 𝐴. Ultimately, this process reduces to the FOL axioms, or their deduction form. If these axioms hold when 𝐴 replaces 𝑥- under the above assumptions - then the replacement can be considered generally valid. The affected FOL axioms are ax-6 1969 (in the form ax6ev 1971), ax-7 2010, ax-8 2116, ax-9 2124, ax-12 2185 (ax12v2 2187), and to some extent ax-13 2376 (ax13v 2377). Since ZF (Zermelo-Fraenkel) set theory does not allow quantifification over class variables, no similar class-based versions of the quantified FOL axioms exist. (Contributed by Wolf Lammen, 18-Sep-2025.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	wl-cleq-6 37817*	Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This text should be read as an exploration rather than as definite statements, open to doubt, alternatives, and reinterpretation. Eliminability of Classes One requirement of Zermelo-Fraenkel set theory (ZF) is that it can be formulated entirely without referring to classes. Since set.mm implements ZF, it must therefore be possible to eliminate all classes from its formalization. Eliminating Variables of Propositional Logic Classical propositional logic concerns statements that are either true or false. For example, "A minute has 60 seconds" is such a statement, as is "English is not a language". Our development of propositional logic applies to all such statements, regardless of their subject matter. Any particular topic, or universe of discourse is encompassed by the general theorems of propositional logic. In ZF, however, the objects of study are sets - mathematical entities. The flexibility of propsitional variables is not required here. Instead, ZF introduces two primitive connectives between sets: 𝑥 = 𝑦 and 𝑥 ∈ 𝑦. ZF is concerned only with logical schemata constructed solely from these primitives. Thus, before we can eliminate classes, we must first eliminate propositional variables like 𝜑 and 𝜓. We will describe this process constructively. We begin by restricting ourselves to propositional schemata that consist only of the primitives of ZF, without any propositional variables. Extending this step to first-order Logic (FOL) - by introducing quantifiers - yields the fundamental predicates of ZF, that is, the basic formulas expressible within it. For convenience, we may again allow propositional variables, but under the strict assumption that they always represent fundamental predicates of ZF. Predicates of level 0 are exactly of this kind: no classes occurs in them, and they can be reduced directly to fundamental predicates in ZF. Introducing eliminable classes The following construction is inspired by a paragraph in Azriel Levy's "Basic set theory" concerning eliminable classes. A class can only occur in combination with one of the operators = or ∈. This applies in particular to class abstractions, which are the only kind of classes permtted in this step of extending level-0 predicates in ZF. The definitions df-cleq 2728 and df-clel 2811 show that equality and membership ultimately reduce to expressions of the form 𝑥 ∈ 𝐴. For a class abstraction {𝑦 ∣ 𝜑}, the resulting term amounts to [𝑥 / 𝑦]𝜑. If 𝜑 is a level-0 predicate, then this too is a level-0 expression - fully compatible with ZF. A level-1 class abstraction is a class {𝑦 ∣ 𝜑} where 𝜑 is a level-0 predicate. A level-1 class abstraction can occur in an equality or membership relation with another level-1 class abstractions or a set variable, and such terms reduce to fundamental predicates. Predicates of either level-0, or containing level-1 class abstractions are called level-1 predicates. After eliminating all level-1 abstractions from such a predicate a level-0 expression is the result. Analoguously, we can define level-2 class abstractions, where the predicate 𝜑 in {𝑦 ∣ 𝜑} is a level-1 predicate. Again, 𝑥 ∈ {𝑦 ∣ 𝜑} reduces to a level-1 expression, which in turn can be reduced to a level-0 one. By similar reasoning, equality and membership between at most level-2 class abstractions also reduce to level-0 expressions. A predicate containing at most level-2 class abstractions is called a level-2 predicate. This iterative construction process can be continued to define a predicate of any level. They can be reduced to fundamental predicates in ZF. Introducing eliminable class variables We have seen that propositional variables must be restricted to representing only primitive connectives to maintain compatibility with ZF. Similarly, class variables can be restricted to representing class abstractions of finite level. Such class variables are eliminable, and even definitions like df-un 3894 (𝐴 ∪ 𝐵) introduce no difficulty, since the resulting union remains of finite level. Limitations of eliminable class variables Where does this construction reach its limits? 1. Infinite constructions. Suppose we wish to add up an infinite series of real numbers, where each term defines its successor using a class abstraction one level higher than that of the previous term. Such a summation introduces terms of arbitrary high level. While each individual term remains reducable in ZF, the infinite sum expression may not be reducable without special care. 2. Class builders. Every class builder other than cv 1541 must be a definition, making its elimination straightforward. The class abstraction df-clab 2715 described above is a special case. Since set variables themselves can be expressed as class abstractions - namely 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} (see cvjust 2730) - this formulation does not conflict with the use of class builder cv 1541. The above conditions apply only to substitution. The expression 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} (abid1 2872) is a valid and provable equation, and it should not be interpreted as an assignment that binds a particular instance to 𝐴. (Contributed by Wolf Lammen, 13-Oct-2025.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Axiom	ax-wl-cleq 37818*	Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This text should be read as an exploration rather than as a definite statement, open to doubt, alternatives, and reinterpretation. At the point where df-cleq 2728 is introduced, the foundations of set theory are being established through the notion of a class. A central property of classes is what elements, expressed by the membership operator ∈, belong to them . Quantification (∀𝑥) applies only to objects that a variable of kind setvar can represent. These objects will henceforth be called sets. Some classes may not be sets; these are called proper classes. It remains open at this stage whether membership can involve them. The formula given in df-cleq 2728 (restated below) asserts that two classes are equal if and only if they have exactly the same sets as elements. If proper classes are also admitted as elements, then two equal classes could still differ by such elements, potentially violating Leibniz's Law. A future axiom df-clel 2811 addresses this issue; df-cleq 2728 alone does not. **Primitive connectives and class builders** Specially crafted primitive operators on classes or class builders could introduce properties of classes beyond membership, not reflected in the formula here. This again risks violating Leibniz's Law. Therefore, the introduction of any future primitive operator or class builder must include a conservativity check to ensure consistency with Leibniz's Law. **This axiom covers only some principles of equality** The notion of equality expressed in this axiom does not automatically coincide with the general notion of equality. Some principles are, however, already captured: Equality is shown to be an equivalence relation, covering transivity (eqtr 2756), reflexivity (eqid 2736) and symmetry (eqcom 2743). It also yields the class-level version of ax-ext 2708 (the backward direction of df-cleq 2728) holds. If we assume 𝑥 = 𝐴 holds, then substituting the free set variable 𝑦 with 𝐴 in ax6ev 1971 and ax12v2 2187 yields provable theorems (see wl-isseteq 37821, and wl-ax12v2cl 37822). However, a bound variable cannot be replaced with a class variable, since quantification over classes is not permitted. Taken together with the results from the previous paragraph, this shows that a class variable equal to a set behaves the same as a set variable, provided it is not quantified. **Conservativity** Moreover, this axiom is already partly derivable if all class variables are replaced by variables of type "setvar". In that case, the statement reduces to an instance of axextb 2711. This shows that the class builder cv 1541 is consistent with this axiom. **Eliminable operator** Finally, this axiom supports the idea that proper classes, and operators between them, should be eliminable, as required by ZF: It reduces equality to their membership properties. However, since the term 𝑥 ∈ 𝐴 is still undefined, elimination reduces equality to just something not yet clarified. **Axiom vs Definition** Up to this point, the only content involving class variables comes from the syntax definitions wceq 1542 and wcel 2114. Axioms are therefore required to progressively refine the semantics of classes until provable results coincide with our intended conception of set theory. This refinement process is explained in Step 4 of wl-cleq-2 37813. From this perspective, df-cleq 2728 is in fact an axiom in disguise and would more appropriately be named ax-cleq. At first glance, one might think that 𝐴 = 𝐵 is defined by the right-hand side of the biconditional. This would make 𝑥 ∈ 𝐴, i.e. membership of a set in a class, the more primitive concept, from which equality of classes could be derived. Such a viewpoint would be coherent if the properties of membership could be fully determined by other axioms. In my (WL's') opinion, however, the more direct and fruitful approach is not to construct class equality from membership, but to treat equality itself as axiomatic. (Contributed by Wolf Lammen, 25-Aug-2025.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Axiom	ax-wl-clel 37819*	Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This text should be read as an exploration rather than as a definite statement, open to doubt, alternatives, and reinterpretation. The formula in df-clel 2811 (restated below) states that only those classes for which ∃𝑥𝑥 = 𝐴 holds can be members of classes. Thus, a member of a class is always equal to a set, which excludes proper classes from class membership. As explained in wl-cleq-4 37815, item 3, ∃𝑥𝑥 = 𝐴 is a sufficient criterion for a class to be a set, provided that Leibniz's Law holds for equality. Therefore this axiom is often rephrased as: classes contain only sets as members. **Principles of equality** Using this axiom we can derive the class-level counterparts of ax-8 2116 (see eleq2 2825) and ax-9 2124 (see eleq1 2824). Since ax-wl-cleq 37818 already asserts that equality between classes is an equivalence relation, the operators = and ∈ alone cannot distinguish equal classes. Hence, if membership is the only property that matters for classes, Leibniz's Law will hold. Later, however, additional class builders may introduce further properties of classes. A conservativity check for such builders can ensure this does not occur. **Eliminability** If we replace the class variable 𝐴 with a set variable 𝑧 in this axiom, the auxiliary variable 𝑥 can be eliminated, leaving only the trivial result (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵). Thus, df-clel 2811 by itself does not determine when a set is a member of a class. From this perspective, df-clel 2811 is in fact an axiom in disguise and would more appropriately be called ax-clel. Overall, our axiomization leaves the meaning of fundamental expressions 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵 open. All other fundamental formulas of set theory (𝐴 not a set variable, 𝐴 ∈ 𝐵, 𝑥 = 𝐵 𝐴 = 𝐵) can be reduced solely to the basic formulas 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵. If an axiomatization leaves a fundamental formula like 𝑥 ∈ 𝐴 unspecified, we could in principle define it bi-conditionally by any formula whatsoever - for example, the trivial ⊤. This, however, is not the approach we take. Instead, an appropriate class builder such as df-clab 2715 fills this gap. (Contributed by Wolf Lammen, 26-Aug-2025.)
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
Theorem	wl-df-clab 37820	Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This text should be seen as an exploration, rather than viewing it as set in stone, no doubt or alternatives possible. We now introduce the notion of class abstraction, which allows us to describe a specific class, in contrast to class variables that can stand for any class indiscriminately. A new syntactic form is introduced for class abstractions, {𝑦 ∣ 𝜑}, read as "the class of sets 𝑦 such that 𝜑(𝑦)". This form is assigned the type "class" in cab 2714, so it can consistently substitute for a class variable during the syntactic construction process. **Eliminability** The axioms ax-wl-cleq 37818 and ax-wl-clel 37819 leave only 𝑥 ∈ 𝐴 unspecified. The definition of this class builder directly corresponds to that expression. When a class abstraction replaces the variable 𝐴 and 𝐵, then 𝐴 = 𝐵 and 𝐴 ∈ 𝐵 can be expressed in terms of these abstractions. For general eliminability two conditions are needed: 1. Any class builder must replace 𝑥 ∈ 𝐴 with an expression containing no class variables. If necessary, class variables must be eliminated via a finite recursive process. 2. There must only be finitely many class builders. If a class variable could range over infinitely many builders, eliminability would fail, since unknown future builders would always need to be considered. Condition (2) is met in set.mm by defining no class builder beyond cv 1541 and df-clab 2715. Thus we may assume that a class variable represents either a set variable, or a class abstraction: a. If it represents a set variable, substitution eliminates it immediately. b. If it equals a set variable 𝑥, then by cvjust 2730 it can be replaced with {𝑦 ∣ 𝑦 ∈ 𝑥}. c. If it represents a proper class, then it equals some abstraction {𝑥 ∣ 𝜑}. If 𝜑 contains no class variables, elimination using 𝜑 is possible. The same holds if finite sequence of elimination steps renders 𝜑 free of class variables. d. It represents a proper class, but 𝜑 in {𝑥 ∣ 𝜑} still contains non-eliminable class variables, then eliminability fails. A simple example is {𝑥 ∣ 𝑥 ∈ 𝐴}. Class variables can only appear in fundamental expressions 𝐴 = 𝐵 or 𝐴 ∈ 𝐵, Both can be reduced to forms involving 𝑧 ∈ 𝐴. Thus, in the expression 𝑧 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴}, we still must eliminate 𝐴. Applying df-clab 2715 reduces it back to 𝑧 ∈ 𝐴, returning us to the starting point. Case (d) shows that in full generality, a class variable cannot always be eliminated, something Zermelo-Fraenkel set theory (ZF) requires. If the universe contained only finitely many sets, a free class variable 𝐴 could be expressed as a finite disjunction of possiblities, hence eliminable. But in ZF's richer universe, in a definition of an unrestricted class variable 𝐴 = {𝑥 ∣ 𝜑} the variable 𝜑 will contain 𝐴 in some way, violating condition (1) above. Thus constraints are needed. In ZF, any formula containing class variables assumes that non-set class variables can be be replaced by {𝑥 ∣ 𝜑} where 𝜑 itself contains no class variables. There is, however, no way to state this condition in a formal way in set.mm. Class abstractions themselves, however, can be eliminated, so df-clab is a definition. **Definition checker** How can case (d) be avoided? A solution is to restrict generality: require that in the definition of any concrete class abstraction {𝑥 ∣ 𝜑}, the formula 𝜑 is either free of class variables or built only from previously defined constructions. Such a restriction could be part of the definition checker. In practice, the Metamath definition checker requires definitions to follow the specific pattern "⊢ {𝑥 ∣ 𝜑} = ...". Although df-clab 2715 does not conform to this pattern, it nevertheless permits elimination of class abstractions. Eliminability is the essential property of a valid definition, so df-clab 2715 can legitimately be regarded as one. For further material on the elimination of class abstractions, see BJ's work beginning with eliminable1 37166 and one comment in https://github.com/metamath/set.mm/pull/4971. (Contributed by Wolf Lammen, 28-Aug-2025.)
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
Theorem	wl-isseteq 37821*	A class equal to a set variable implies it is a set. Note that 𝐴 may be dependent on 𝑥. The consequent, resembling ax6ev 1971, is the accepted expression for the idea of a class being a set. Sometimes a simpler expression like the antecedent here, or in elisset 2818, is already sufficient to mark a class variable as a set. (Contributed by Wolf Lammen, 7-Sep-2025.)
⊢ (𝑥 = 𝐴 → ∃𝑦 𝑦 = 𝐴)
Theorem	wl-ax12v2cl 37822*	The class version of ax12v2 2187, where the set variable 𝑦 is replaced with the class variable 𝐴. This is possible if 𝐴 is known to be a set, expressed by the antecedent. Theorem ax12v 2186 is a specialization of ax12v2 2187. So any proof using ax12v 2186 will still hold if ax12v2 2187 is used instead. Theorem ax12v2 2187 expresses that two equal set variables cannot be distinguished by whatever complicated formula 𝜑 if one is replaced with the other in it. This theorem states a similar result for a class variable known to be a set: All sets equal to the class variable behave the same if they replace the class variable in 𝜑. Most axioms in FOL containing an equation correspond to a theorem where a class variable known to be a set replaces a set variable in the formula. Some exceptions cannot be avoided: The set variable must nowhere be bound. And it is not possible to state a distinct variable condition where a class 𝐴 is different from another, or distinct from a variable with type wff. So ax-12 2185 proper is out of reach: you cannot replace 𝑦 in ∀𝑦𝜑 with a class variable. But where such limitations are not violated, the proof of the FOL theorem should carry over to a version where a class variable, known to be set, appears instead of a set variable. (Contributed by Wolf Lammen, 8-Aug-2020.)
⊢ (∃𝑦 𝑦 = 𝐴 → (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
Theorem	wl-df.clab 37823	Define class abstractions, that is, classes of the form {𝑦 ∣ 𝜑}, which is read "the class of sets 𝑦 such that 𝜑(𝑦)". A few remarks are in order: 1. The axiomatic statement df-clab 2715 does not define the class abstraction {𝑦 ∣ 𝜑} itself, that is, it does not have the form ⊢ {𝑦 ∣ 𝜑} = ... that a standard definition should have (for a good reason: equality itself has not yet been defined or axiomatized for class abstractions; it is defined later in df-cleq 2728). Instead, df-clab 2715 has the form ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ...), meaning that it only defines what it means for a setvar to be a member of a class abstraction. As a consequence, one can say that df-clab 2715 defines class abstractions if and only if a class abstraction is completely determined by which elements belong to it, which is the content of the axiom of extensionality ax-ext 2708. Therefore, df-clab 2715 can be considered a definition only in systems that can prove ax-ext 2708 (and the necessary first-order logic). 2. As in all definitions, the definiendum (the left-hand side of the biconditional) has no disjoint variable conditions. In particular, the setvar variables 𝑥 and 𝑦 need not be distinct, and the formula 𝜑 may depend on both 𝑥 and 𝑦. This is necessary, as with all definitions, since if there was for instance a disjoint variable condition on 𝑥, 𝑦, then one could not do anything with expressions like 𝑥 ∈ {𝑥 ∣ 𝜑} which are sometimes useful to shorten proofs (because of abid 2718). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑. 3. Remark 1 stresses that df-clab 2715 does not have the standard form of a definition for a class, but one could be led to think it has the standard form of a definition for a formula. However, it also fails that test since the membership predicate ∈ has already appeared earlier (outside of syntax e.g. in ax-8 2116). Indeed, the definiendum extends, or "overloads", the membership predicate ∈ from formulas of the form "setvar ∈ setvar" to formulas of the form "setvar ∈ class abstraction". This is possible because of wcel 2114 and cab 2714, and it can be called an "extension" of the membership predicate because of wel 2115, whose proof uses cv 1541. An a posteriori justification for cv 1541 is given by cvjust 2730, stating that every setvar can be written as a class abstraction (though conversely not every class abstraction is a set, as illustrated by Russell's paradox ru 3726). 4. Proof techniques. Because class variables can be substituted with compound expressions and setvar variables cannot, it is often useful to convert a theorem containing a free setvar variable to a more general version with a class variable. This is done with theorems such as vtoclg 3499 which is used, for example, to convert elirrv 9512 to elirr 9514. 5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2715, df-cleq 2728, and df-clel 2811, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 30470), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2728 and df-clel 2811). One could be less strict and decide to call "definition" every axiomatic statement which provides an eliminable and conservative extension of the considered axiom system. But the notion of conservativity may be given two different meanings in set.mm, due to the difference between the "scheme level" of set.mm and the "object level" of classical treatments. For a proof that these three axiomatic statements yield an eliminable and weakly (that is, object-level) conservative extension of FOL= plus ax-ext 2708, see Appendix of [Levy] p. 357. 6. This definition (or axiom) is a class builder introducing the class {𝑥 ∣ 𝜑}, also called a class abstraction or class comprehension, i.e., it specifies a class by a condition determining its members. Another class-building operator (but no class abstraction) is cv 1541, which asserts that every set is a class. The converse need not hold: not every class is a set. A class that is not a set is called a proper class. From ru 3726, it follows that this abstraction yields proper classes, e.g. {𝑥 ∣ 𝑥 = 𝑥}. 7. References and history. The concept of class abstraction dates back to at least Frege, and is used by Whitehead and Russell. This definition is Definition 2.1 of [Quine] p. 16 and Axiom 4.3.1 of [Levy] p. 12. It is called the "axiom of class comprehension" by [Levy] p. 358, who treats the theory of classes as an extralogical extension to predicate logic and set theory axioms. He calls the construction {𝑦 ∣ 𝜑} a "class term". For a full description of how classes are introduced and how to recover the primitive language, see the books of Quine and Levy (and the comment of eqabb 2875 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2875. (Contributed by NM, 26-May-1993.) (Revised by BJ, 19-Aug-2023.)
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
Theorem	wl-df.cleq 37824*	Define the equality connective between classes. Definition 2.7 of [Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4 provides its justification and methods for eliminating it. Note that its elimination will not necessarily result in a single wff in the original language but possibly a "scheme" of wffs. The hypotheses express that all instances of the conclusion where class variables are replaced with setvar variables hold. Therefore, this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This is only a proof sketch of conservativity; for details see Appendix of [Levy] p. 357. This is the reason why we call this axiomatic statement a "definition", even though it does not have the usual form of a definition. If we required a definition to have the usual form, we would call df-cleq 2728 an axiom. See also comments under df-clab 2715, df-clel 2811, and eqabb 2875. In the form of dfcleq 2729, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. It characterizes classes as collections of sets. While the three class definitions df-clab 2715, df-cleq 2728, and df-clel 2811 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker. For a general discussion of the theory of classes, see mmset.html#class 2811. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.)
⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧))    &   ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡))    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	wl-dfcleq.basic 37825*	This theorem is a conservative extension of ax-ext 2708 to classes, with no hypotheses. It is not complete, since ax-8 2116 can be derived (see in-ax8 36406) via alpha-renaming. Although unsuitable for general use, it is adequate for the development of theorems unaffected by alpha-renaming, including: 1. Theorems with no bound variables in the hypotheses or conclusion (see eqriv 2733). 2. Theorems using the same bound variable throughout (see abbib 2805). 3. Theorems with distinct bound variables arising only through implicit substitution (see eqabbw 2809). Remark: the proof uses axextb 2711 to prove the hypothesis of df-cleq 2728 that is a degenerate instance, but it could be proved also from minimal propositional calculus and { ax-gen 1797, equid 2014 }. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	wl-dfcleq.just 37826*	The hypotheses added to this version of df-cleq 2728 address the following: 1. Equality of classes is an equivalence relation, as expected of equality. 2. Equality of classes obeys the Law of Indiscernibles (Leibniz's Law), and is compatible with class membership. 3. Alpha-renaming is explicitly permitted. (Contributed by Wolf Lammen, 7-Apr-2026.)
⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))    &   ⊢ 𝐴 = 𝐴    &   ⊢ (𝐴 = 𝐵 → (𝐵 = 𝐶 → 𝐶 = 𝐴))    &   ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 → 𝐵 ∈ 𝐶))    &   ⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	wl-df.clel 37827*	Define the membership connective between classes. Theorem 6.3 of [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we adopt as a definition. See these references for its metalogical justification. The hypotheses assert that every instance of the conclusion obtained by substituting the class variables with set variables already holds. Thus, this definition extends to class variables a relation already valid for set variables, and is therefore conservative. This only sketches the conservativity arguement; for details see Appendix of [Levy] p. 357. For this reason we regard this statement as a "definition", even though it does not have the usual form of a definition. Under a stricter syntactic criterion, df-clel 2811 would instead be an axiom. See also comments under df-clab 2715, df-cleq 2728, and eqabb 2875. Alternate characterizations of 𝐴 ∈ 𝐵 when either 𝐴 or 𝐵 is a set are given by clel2g 3601, clel3g 3603, and clel4g 3605. [Levy] p. 338 refers to this as the "axiom of membership", treating the theory of classes as an extralogical extension to our logic and set theory axioms. Under this definition, class members can only be sets; classes are therefore collections of sets. Although the extensionality expressed in df-cleq 2728 already points in this direction, an unusual interpretation of equality could still permit proper classes as members. Although the class definitions df-clab 2715, df-cleq 2728, and df-clel 2811 are eliminable and conservative, and hence meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the syntactic requirements enforced by the current definition checker. The conservativity proofs require external justification beyond the scope of the checker. For a general discussion of the theory of classes, see mmset.html#class 2811. (Contributed by NM, 26-May-1993.) (Revised by BJ, 27-Jun-2019.)
⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧))    &   ⊢ (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡))    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
Theorem	wl-dfclel.basic 37828*	This theorem gives a conservative extension of membership of classes, without hypotheses. Conservativity alone, however, is insufficient, since issues involving alpha-renaming can still arise, see in-ax8 36406. Although unsuitable for general use, it is adequate for the development of theorems unaffected by alpha-renaming, including: 1. Theorems whose hypotheses and conclusion contain no bound variables (see eleq1w 2819). 2. Theorems using the same bound variable throughout (see elex2 2813). 3. Theorems in which distinct bound variables arise only through implicit substitution (see eqabbw 2809). (Contributed by BJ, 27-Jun-2019.)
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
Theorem	wl-dfclel.just 37829*	Add a hypothesis to wl-dfclel.basic 37828, that permits alpha-renaming. (Contributed by Wolf Lammen, 7-Apr-2026.)
⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵))    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
Theorem	wl-dfcleq 37830*	The defining characterization of class equality. This version of df-cleq 2728 has no restrictions, unlike the forms on which it is based. It is proved in Tarski's FOL from the axiom of extensionality (ax-ext 2708), the definition of class equality (df-cleq 2728), and the definition of class membership (df-clel 2811). Its forward implication is known as "class extensionality". (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.) Base on wl-dfcleq.just 37826. (Revised by Wolf Lammen, 7-Apr-2026.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	wl-dfclel 37831*	The defining characterization of class membership. Unlike the forms on which it is based, it is unrestricted. Proven in Tarski's FOL, from the axiom of (set) extensionality (ax-ext 2708), the definitions df-clel 2811 and df-cleq . (Contributed by BJ, 27-Jun-2019.) Base on wl-dfclel.just 37829. (Revised by Wolf Lammen, 13-Apr-2026.)
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
21.22.7  Other stuff
Theorem	wl-mps 37832	Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ ((𝜑 → 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜃)
Theorem	wl-syls1 37833	Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜓 → 𝜒)    &   ⊢ ((𝜑 → 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜃)
Theorem	wl-syls2 37834	Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜑 → 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 → 𝜒) → 𝜃)
Theorem	wl-embant 37835	A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜒)
Theorem	wl-orel12 37836	In a conjunctive normal form a pair of nodes like (𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒) eliminates the need of a node (𝜓 ∨ 𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.)
⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜒))
Theorem	wl-cases2-dnf 37837	A particular instance of orddi 1012 and anddi 1013 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 1048, and is related to consensus 1053. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1064 and dfifp4 1067, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.)
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
Theorem	wl-cbvmotv 37838*	Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.)
⊢ (∃*𝑥⊤ → ∃*𝑦⊤)
Theorem	wl-moteq 37839	Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.)
⊢ (∃*𝑥⊤ → 𝑦 = 𝑧)
Theorem	wl-motae 37840	Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.)
⊢ (∃*𝑢⊤ → ∀𝑥 𝑦 = 𝑧)
Theorem	wl-moae 37841*	Two ways to express "at most one thing exists" or, in this context equivalently, "exactly one thing exists" . The equivalence results from the presence of ax-6 1969 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 37842 and exists1 2661. Gerard Lang pointed out, that ∃𝑦∀𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (dfmo 2540, trut 1548) also means "exactly one thing exists" . (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant ⊤. (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies, and use ∃*. (Revised by Wolf Lammen, 7-Mar-2023.)
⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
Theorem	wl-euae 37842*	Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.)
⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
Theorem	wl-nax6im 37843*	The following series of theorems are centered around the empty domain, where no set exists. As a consequence, a set variable like 𝑥 has no instance to assign to. An expression like 𝑥 = 𝑦 is not really meaningful then. What does it evaluate to, true or false? In fact, the grammar extension weq 1964 requires us to formally assign a boolean value to an equation, say always false, unless you want to give up on exmid 895, for example. Whatever it is, we start out with the contraposition of ax-6 1969, that guarantees the existence of at least one set. Our hypothesis here expresses tentatively it might not hold. We can simplify the antecedent then, to the point where we do not need equation any more. This suggests what a decent characterization of the empty domain of discourse could be. (Contributed by Wolf Lammen, 12-Mar-2023.)
⊢ (¬ ∃𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (¬ ∃𝑥⊤ → 𝜑)
Theorem	wl-hbae1 37844	This specialization of hbae 2435 does not depend on ax-11 2163. (Contributed by Wolf Lammen, 8-Aug-2021.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)
Theorem	wl-naevhba1v 37845*	An instance of hbn1w 2050 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	wl-spae 37846	Prove an instance of sp 2191 from ax-13 2376 and Tarski's FOL only, without distinct variable conditions. The antecedent ∀𝑥𝑥 = 𝑦 holds in a multi-object universe only if 𝑦 is substituted for 𝑥, or vice versa, i.e. both variables are effectively the same. The converse ¬ ∀𝑥𝑥 = 𝑦 indicates that both variables are distinct, and it so provides a simple translation of a distinct variable condition to a logical term. In case studies ∀𝑥𝑥 = 𝑦 and ¬ ∀𝑥𝑥 = 𝑦 can help eliminating distinct variable conditions. The antecedent ∀𝑥𝑥 = 𝑦 is expressed in the theorem's name by the abbreviation ae standing for 'all equal'. Note that we cannot provide a logical predicate telling us directly whether a logical expression contains a particular variable, as such a construct would usually contradict ax-12 2185. Note that this theorem is also provable from ax-12 2185 alone, so you can pick the axiom it is based on. Compare this result to 19.3v 1984 and spaev 2056 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
Theorem	wl-speqv 37847*	Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2191 is provable from Tarski's FOL and ax13v 2377 only. Note that this reverts the implication in ax13lem1 2378, so in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
⊢ (¬ 𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
Theorem	wl-19.8eqv 37848*	Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 2189 is provable from Tarski's FOL and ax13v 2377 only. Note that this reverts the implication in ax13lem2 2380, so in fact (¬ 𝑥 = 𝑦 → (∃𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦))
Theorem	wl-19.2reqv 37849*	Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1978 is provable from Tarski's FOL and ax13v 2377 only. Note that in conjunction with 19.2 1978 in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Theorem	wl-nfalv 37850*	If 𝑥 is not present in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Wolf Lammen, 11-Jan-2020.)
⊢ Ⅎ𝑥∀𝑦𝜑
Theorem	wl-nfimf1 37851	An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1898 in dvelimdf 2453 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.)
⊢ (∀𝑥𝜑 → (Ⅎ𝑥(𝜑 → 𝜓) ↔ Ⅎ𝑥𝜓))
Theorem	wl-nfae1 37852	Unlike nfae 2437, this specialized theorem avoids ax-11 2163. (Contributed by Wolf Lammen, 26-Jun-2019.)
⊢ Ⅎ𝑥∀𝑦 𝑦 = 𝑥
Theorem	wl-nfnae1 37853	Unlike nfnae 2438, this specialized theorem avoids ax-11 2163. (Contributed by Wolf Lammen, 27-Jun-2019.)
⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥
Theorem	wl-aetr 37854	A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧))
Theorem	wl-axc11r 37855	Same as axc11r 2372, but using ax12 2427 instead of ax-12 2185 directly. This better reflects axiom usage in theorems dependent on it. (Contributed by NM, 25-Jul-2015.) Avoid direct use of ax-12 2185. (Revised by Wolf Lammen, 30-Mar-2024.)
⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
Theorem	wl-dral1d 37856	A version of dral1 2443 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 𝑥 = 𝑦, ∀𝑥𝑥 = 𝑦 or 𝜑 antecedent. wl-equsal1i 37869 and nf5di 2292 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))
Theorem	wl-cbvalnaed 37857	wl-cbvalnae 37858 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓))    &   ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	wl-cbvalnae 37858	A more general version of cbval 2402 when nonfree properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2452, nfsb2 2487 or dveeq1 2384. (Contributed by Wolf Lammen, 4-Jun-2019.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	wl-exeq 37859	The semantics of ∃𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
⊢ (∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
Theorem	wl-aleq 37860	The semantics of ∀𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
⊢ (∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))
Theorem	wl-nfeqfb 37861	Extend nfeqf 2385 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.)
⊢ (Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
Theorem	wl-nfs1t 37862	If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2492. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	wl-equsalvw 37863*	Version of equsalv 2275 with a disjoint variable condition, and of equsal 2421 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2007. This theorem lays the foundation to a transformation of expressions called substitution of set variables in a wff. Only in this particular context we additionally assume 𝜑 and 𝑦 disjoint, stated here as 𝜑(𝑥). Similarly the disjointness of 𝜓 and 𝑥 is expressed by 𝜓(𝑦). Both 𝜑 and 𝜓 may still depend on other set variables, but that is irrelevant here. We want to transform 𝜑(𝑥) into 𝜓(𝑦) such that 𝜓 depends on 𝑦 the same way as 𝜑 depends on 𝑥. This dependency is expressed in our hypothesis (called implicit substitution): (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)). For primitive enough 𝜑 a sort of textual substitution of 𝑥 by 𝑦 is sufficient for such transformation. But note: 𝜑 must not contain wff variables, and the substitution is no proper textual substitution either. We still need grammar information to not accidently replace the x in a token 'x.' denoting multiplication, but only catch set variables 𝑥. Our current stage of development allows only equations and quantifiers make up such primitives. Thanks to equequ1 2027 and cbvalvw 2038 we can then prove in a mechanical way that in fact the implicit substitution holds for each instance. If 𝜑 contains wff variables we cannot use textual transformation any longer, since we don't know how to replace 𝑦 for 𝑥 in placeholders of unknown structure. Our theorem now states, that the generic expression ∀𝑥(𝑥 = 𝑦 → 𝜑) formally behaves as if such a substitution was possible and made. (Contributed by BJ, 31-May-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	wl-equsald 37864	Deduction version of equsal 2421. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
Theorem	wl-equsaldv 37865*	Deduction version of equsal 2421. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
Theorem	wl-equsal 37866	A useful equivalence related to substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) It seems proving wl-equsald 37864 first, and then deriving more specialized versions wl-equsal 37866 and wl-equsal1t 37867 then is more efficient than the other way round, which is possible, too. See also equsal 2421. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	wl-equsal1t 37867	The expression 𝑥 = 𝑦 in antecedent position plays an important role in predicate logic, namely in implicit substitution. However, occasionally it is irrelevant, and can safely be dropped. A sufficient condition for this is when 𝑥 (or 𝑦 or both) is not free in 𝜑. This theorem is more fundamental than equsal 2421, spimt 2390 or sbft 2277, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	wl-equsalcom 37868	This simple equivalence eases substitution of one expression for the other. (Contributed by Wolf Lammen, 1-Sep-2018.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑦 = 𝑥 → 𝜑))
Theorem	wl-equsal1i 37869	The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.)
⊢ (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)    &   ⊢ (𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	wl-sbid2ft 37870*	A more general version of sbid2vw 2267. (Contributed by Wolf Lammen, 14-May-2019.)
⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑))
Theorem	wl-cbvalsbi 37871*	Change bounded variables in a special case. The reverse direction seems to involve ax-11 2163. My hope is that I will in some future be able to prove mo3 2564 with reversed quantifiers not using ax-11 2163. See also the remark in mo4 2566, which lead me to this effort. (Contributed by Wolf Lammen, 5-Mar-2024.)
⊢ (∀𝑥𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	wl-sbrimt 37872	Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2311. (Contributed by Wolf Lammen, 26-Jul-2019.)
⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))
Theorem	wl-sblimt 37873	Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2311. (Contributed by Wolf Lammen, 26-Jul-2019.)
⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)))
Theorem	wl-sb9v 37874*	Commutation of quantification and substitution variables based on fewer axioms than sb9 2523. (Contributed by Wolf Lammen, 27-Apr-2025.)
⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	wl-sb8ft 37875*	Substitution of variable in universal quantifier. Closed form of sb8f 2358. (Contributed by Wolf Lammen, 27-Apr-2025.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8eft 37876*	Substitution of variable in existentialal quantifier. Closed form of sb8ef 2359. (Contributed by Wolf Lammen, 27-Apr-2025.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8t 37877	Substitution of variable in universal quantifier. Closed form of sb8 2521. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8et 37878	Substitution of variable in universal quantifier. Closed form of sb8e 2522. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sbhbt 37879	Closed form of sbhb 2525. Characterizing the expression 𝜑 → ∀𝑥𝜑 using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
Theorem	wl-sbnf1 37880	Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2362. Note: This theorem shows that sbnf2 2362 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
Theorem	wl-equsb3 37881	equsb3 2109 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.)
⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧))
Theorem	wl-equsb4 37882	Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable condition. (Contributed by Wolf Lammen, 26-Jun-2019.)
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
Theorem	wl-2sb6d 37883	Version of 2sb6 2092 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑥)    &   ⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑤)    &   ⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑧)    &   ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)    ⇒   ⊢ (𝜑 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜓)))
Theorem	wl-sbcom2d-lem1 37884*	Lemma used to prove wl-sbcom2d 37886. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
Theorem	wl-sbcom2d-lem2 37885*	Lemma used to prove wl-sbcom2d 37886. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝜑)))
Theorem	wl-sbcom2d 37886	Version of sbcom2 2179 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑤)    &   ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)    &   ⊢ (𝜑 → ¬ ∀𝑧 𝑧 = 𝑦)    ⇒   ⊢ (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))
Theorem	wl-sbalnae 37887	A theorem used in elimination of disjoint variable restrictions by replacing them with distinctors. (Contributed by Wolf Lammen, 25-Jul-2019.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Theorem	wl-sbal1 37888*	A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) Proof is based on wl-sbalnae 37887 now. See also sbal1 2532. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Theorem	wl-sbal2 37889*	Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002.) Proof is based on wl-sbalnae 37887 now. See also sbal2 2533. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Theorem	wl-2spsbbi 37890	spsbbi 2079 applied twice. (Contributed by Wolf Lammen, 5-Aug-2023.)
⊢ (∀𝑎∀𝑏(𝜑 ↔ 𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))
Theorem	wl-lem-exsb 37891*	This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	wl-lem-nexmo 37892	This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑧))
Theorem	wl-lem-moexsb 37893*	The antecedent ∀𝑥(𝜑 → 𝑥 = 𝑧) relates to ∃*𝑥𝜑, but is better suited for usage in proofs. Note that no distinct variable restriction is placed on 𝜑. This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))
Theorem	wl-alanbii 37894	This theorem extends alanimi 1818 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))
Theorem	wl-mo2df 37895	Version of mof 2563 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 11-Aug-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    ⇒   ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)))
Theorem	wl-mo2tf 37896	Closed form of mof 2563 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 20-Sep-2020.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	wl-eudf 37897	Version of eu6 2574 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    ⇒   ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦)))
Theorem	wl-eutf 37898	Closed form of eu6 2574 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
Theorem	wl-euequf 37899	euequ 2597 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)
Theorem	wl-mo2t 37900*	Closed form of mof 2563. (Contributed by Wolf Lammen, 18-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))