P. 376 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37501-37600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wl-cleq-5 37501*	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 1967, ax-7 2007, ax-8 2110, ax-9 2118 and ax-12 2177, and ax-13 2377. In practice, restricted versions with distinct variable conditions are used (ax6v 1968, ax12v 2178). The unrestricted forms together with axiom ax-13 2377, 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 2007, with some support of ax-6 1967. 2b. Leibniz's Law for the primitive ∈ operator. Captured by ax-8 2110 and ax-9 2118. 2c. General formulation. Given in sbequ12 2251. 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 2035). The auxiliary axioms ax-10 2141, ax-11 2157, ax-12 are provable (see ax10w 2129, 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 1539) 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 2729, 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 2816 reduces possible membership relations between class variables to those between a set variable and a class variable. Axiom df-cleq 2729 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 1539, 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 2729 and df-clel 2816 provide class versions of ax-8 2110 and ax-9 2118, 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 1967 (in the form ax6ev 1969), ax-7 2007, ax-8 2110, ax-9 2118, ax-12 2177 (ax12v2 2179), and to some extent ax-13 2377 (ax13v 2378). 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 37502*	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 2729 and df-clel 2816 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 3956 (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 1539 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 2731) - this formulation does not conflict with the use of class builder cv 1539. The above conditions apply only to substitution. The expression 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} (abid1 2878) 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 37503*	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 2729 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 2729 (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 2816 addresses this issue; df-cleq 2729 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 2760), reflexivity (eqid 2737) and symmetry (eqcom 2744). It also yields the class-level version of ax-ext 2708 (the backward direction of df-cleq 2729) holds. If we assume 𝑥 = 𝐴 holds, then substituting the free set variable 𝑦 with 𝐴 in ax6ev 1969 and ax12v2 2179 yields provable theorems (see wl-isseteq 37506, and wl-ax12v2cl 37507). 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 1539 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 1540 and wcel 2108. 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 37498. From this perspective, df-cleq 2729 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 37504*	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 2816 (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 37500, 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 2110 (see eleq2 2830) and ax-9 2118 (see eleq1 2829). Since ax-wl-cleq 37503 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 2816 by itself does not determine when a set is a member of a class. From this perspective, df-clel 2816 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 37505	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 37503 and ax-wl-clel 37504 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 1539 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 2731 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 36860 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 37506*	A class equal to a set variable implies it is a set. Note that 𝐴 may be dependent on 𝑥. The consequent, resembling ax6ev 1969, is the accepted expression for the idea of a class being a set. Sometimes a simpler expression like the antecedent here, or in elisset 2823, 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 37507*	The class version of ax12v2 2179, 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 2178 is a specialization of ax12v2 2179. So any proof using ax12v 2178 will still hold if ax12v2 2179 is used instead. Theorem ax12v2 2179 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 2177 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))))
21.22.7  Other stuff
Theorem	wl-mps 37508	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 37509	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 37510	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 37511	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 37512	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 37513	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 37514*	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 37515	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 37516	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 37517*	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 1967 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 37518 and exists1 2661. Gerard Lang pointed out, that ∃𝑦∀𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2540, trut 1546) 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 37518*	Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.)
⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
Theorem	wl-nax6im 37519*	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 1962 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 1967, 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 37520	This specialization of hbae 2436 does not depend on ax-11 2157. (Contributed by Wolf Lammen, 8-Aug-2021.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)
Theorem	wl-naevhba1v 37521*	An instance of hbn1w 2046 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	wl-spae 37522	Prove an instance of sp 2183 from ax-13 2377 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 2177. Note that this theorem is also provable from ax-12 2177 alone, so you can pick the axiom it is based on. Compare this result to 19.3v 1981 and spaev 2052 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 37523*	Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2183 is provable from Tarski's FOL and ax13v 2378 only. Note that this reverts the implication in ax13lem1 2379, 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 37524*	Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 2181 is provable from Tarski's FOL and ax13v 2378 only. Note that this reverts the implication in ax13lem2 2381, 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 37525*	Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1976 is provable from Tarski's FOL and ax13v 2378 only. Note that in conjunction with 19.2 1976 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 37526*	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 37527	An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1896 in dvelimdf 2454 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.)
⊢ (∀𝑥𝜑 → (Ⅎ𝑥(𝜑 → 𝜓) ↔ Ⅎ𝑥𝜓))
Theorem	wl-nfae1 37528	Unlike nfae 2438, this specialized theorem avoids ax-11 2157. (Contributed by Wolf Lammen, 26-Jun-2019.)
⊢ Ⅎ𝑥∀𝑦 𝑦 = 𝑥
Theorem	wl-nfnae1 37529	Unlike nfnae 2439, this specialized theorem avoids ax-11 2157. (Contributed by Wolf Lammen, 27-Jun-2019.)
⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥
Theorem	wl-aetr 37530	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 37531	Same as axc11r 2371, but using ax12 2428 instead of ax-12 2177 directly. This better reflects axiom usage in theorems dependent on it. (Contributed by NM, 25-Jul-2015.) Avoid direct use of ax-12 2177. (Revised by Wolf Lammen, 30-Mar-2024.)
⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
Theorem	wl-dral1d 37532	A version of dral1 2444 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 37545 and nf5di 2285 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 37533	wl-cbvalnae 37534 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 37534	A more general version of cbval 2403 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 2453, nfsb2 2488 or dveeq1 2385. (Contributed by Wolf Lammen, 4-Jun-2019.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	wl-exeq 37535	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 37536	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 37537	Extend nfeqf 2386 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.)
⊢ (Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
Theorem	wl-nfs1t 37538	If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2493. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	wl-equsalvw 37539*	Version of equsalv 2267 with a disjoint variable condition, and of equsal 2422 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2004. 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 2024 and cbvalvw 2035 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 37540	Deduction version of equsal 2422. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
Theorem	wl-equsaldv 37541*	Deduction version of equsal 2422. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
Theorem	wl-equsal 37542	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 37540 first, and then deriving more specialized versions wl-equsal 37542 and wl-equsal1t 37543 then is more efficient than the other way round, which is possible, too. See also equsal 2422. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	wl-equsal1t 37543	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 2422, spimt 2391 or sbft 2270, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	wl-equsalcom 37544	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 37545	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 37546*	A more general version of sbid2vw 2259. (Contributed by Wolf Lammen, 14-May-2019.)
⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑))
Theorem	wl-cbvalsbi 37547*	Change bounded variables in a special case. The reverse direction seems to involve ax-11 2157. My hope is that I will in some future be able to prove mo3 2564 with reversed quantifiers not using ax-11 2157. 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 37548	Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2304. (Contributed by Wolf Lammen, 26-Jul-2019.)
⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))
Theorem	wl-sblimt 37549	Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2304. (Contributed by Wolf Lammen, 26-Jul-2019.)
⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)))
Theorem	wl-sb9v 37550*	Commutation of quantification and substitution variables based on fewer axioms than sb9 2524. (Contributed by Wolf Lammen, 27-Apr-2025.)
⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	wl-sb8ft 37551*	Substitution of variable in universal quantifier. Closed form of sb8f 2356. (Contributed by Wolf Lammen, 27-Apr-2025.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8eft 37552*	Substitution of variable in existentialal quantifier. Closed form of sb8ef 2358. (Contributed by Wolf Lammen, 27-Apr-2025.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8t 37553	Substitution of variable in universal quantifier. Closed form of sb8 2522. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8et 37554	Substitution of variable in universal quantifier. Closed form of sb8e 2523. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sbhbt 37555	Closed form of sbhb 2526. 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 37556	Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2361. Note: This theorem shows that sbnf2 2361 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
Theorem	wl-equsb3 37557	equsb3 2103 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.)
⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧))
Theorem	wl-equsb4 37558	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 37559	Version of 2sb6 2086 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 37560*	Lemma used to prove wl-sbcom2d 37562. (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 37561*	Lemma used to prove wl-sbcom2d 37562. (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 37562	Version of sbcom2 2173 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 37563	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 37564*	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 37563 now. See also sbal1 2533. (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 37565*	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 37563 now. See also sbal2 2534. (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 37566	spsbbi 2073 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 37567*	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 37568	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 37569*	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 37570	This theorem extends alanimi 1816 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 37571	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 37572	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 37573	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 37574	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 37575	euequ 2597 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)
Theorem	wl-mo2t 37576*	Closed form of mof 2563. (Contributed by Wolf Lammen, 18-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	wl-mo3t 37577*	Closed form of mo3 2564. (Contributed by Wolf Lammen, 18-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Theorem	wl-nfsbtv 37578*	Closed form of nfsbv 2330. (Contributed by Wolf Lammen, 2-May-2025.)
⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
Theorem	wl-sb8eut 37579	Substitution of variable in universal quantifier. Closed form of sb8eu 2600. (Contributed by Wolf Lammen, 11-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8eutv 37580*	Substitution of variable in universal quantifier. Closed form of sb8euv 2599. (Contributed by Wolf Lammen, 3-May-2025.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8mot 37581	Substitution of variable in universal quantifier. Closed form of sb8mo 2601. (Contributed by Wolf Lammen, 11-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8motv 37582*	Substitution of variable in universal quantifier. Closed form of sb8mo 2601 without ax-13 2377, but requiring 𝑥 and 𝑦 being disjoint. This theorem relates to wl-mo3t 37577, since replacing 𝜑 with [𝑦 / 𝑥]𝜑 in the latter yields subexpressions like [𝑥 / 𝑦][𝑦 / 𝑥]𝜑, which can be reduced to 𝜑 via sbft 2270 and sbco 2512. So ∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑 is provable from wl-mo3t 37577 in a simple fashion. From an educational standpoint, one would assume wl-mo3t 37577 to be more fundamental, as it hints how the "at most one" objects on both sides of the biconditional correlate (they are the same), if they exist at all, and then prove this theorem from it. (Contributed by Wolf Lammen, 3-May-2025.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-issetft 37583	A closed form of issetf 3497. The proof here is a modification of a subproof in vtoclgft 3552, where it could be used to shorten the proof. (Contributed by Wolf Lammen, 25-Jan-2025.)
⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))
Theorem	wl-axc11rc11 37584	Proving axc11r 2371 from axc11 2435. The hypotheses are two instances of axc11 2435 used in the proof here. Some systems introduce axc11 2435 as an axiom, see for example System S2 in https://us.metamath.org/downloads/finiteaxiom.pdf 2435. By contrast, this database sees the variant axc11r 2371, directly derived from ax-12 2177, as foundational. Later axc11 2435 is proven somewhat trickily, requiring ax-10 2141 and ax-13 2377, see its proof. (Contributed by Wolf Lammen, 18-Jul-2023.)
⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))    &   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
Axiom	ax-wl-11v 37585*	Version of ax-11 2157 with distinct variable conditions. Currently implemented as an axiom to detect unintended references to the foundational axiom ax-11 2157. It will later be converted into a theorem directly based on ax-11 2157. (Contributed by Wolf Lammen, 28-Jun-2019.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	wl-ax11-lem1 37586	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-ax11-lem2 37587*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)
Theorem	wl-ax11-lem3 37588*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦)
Theorem	wl-ax11-lem4 37589*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ Ⅎ𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	wl-ax11-lem5 37590	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
Theorem	wl-ax11-lem6 37591*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑))
Theorem	wl-ax11-lem7 37592	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))
Theorem	wl-ax11-lem8 37593*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥𝜑))
Theorem	wl-ax11-lem9 37594	The easy part when 𝑥 coincides with 𝑦. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑))
Theorem	wl-ax11-lem10 37595*	We now have prepared everything. The unwanted variable 𝑢 is just in one place left. pm2.61 192 can be used in conjunction with wl-ax11-lem9 37594 to eliminate the second antecedent. Missing is something along the lines of ax-6 1967, so we could remove the first antecedent. But the Metamath axioms cannot accomplish this. Such a rule must reside one abstraction level higher than all others: It says that a distinctor implies a distinct variable condition on its contained setvar. This is only needed if such conditions are required, as ax-11v does. The result of this study is for me, that you cannot introduce a setvar capturing this condition, and hope to eliminate it later. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑦 𝑦 = 𝑢 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)))
Theorem	wl-clabv 37596*	Variant of df-clab 2715, where the element 𝑥 is required to be disjoint from the class it is taken from. This restriction meets similar ones found in other definitions and axioms like ax-ext 2708, df-clel 2816 and df-cleq 2729. 𝑥 ∈ 𝐴 with 𝐴 depending on 𝑥 can be the source of side effects, that you rather want to be aware of. So here we eliminate one possible way of letting this slip in instead. An expression 𝑥 ∈ 𝐴 with 𝑥, 𝐴 not disjoint, is now only introduced either via ax-8 2110, ax-9 2118, or df-clel 2816. Theorem cleljust 2117 shows that a possible choice does not matter. The original df-clab 2715 can be rederived, see wl-dfclab 37597. In an implementation this theorem is the only user of df-clab. (Contributed by NM, 26-May-1993.) Element and class are disjoint. (Revised by Wolf Lammen, 31-May-2023.)
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
Theorem	wl-dfclab 37597	Rederive df-clab 2715 from wl-clabv 37596. (Contributed by Wolf Lammen, 31-May-2023.) (Proof modification is discouraged.)
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
Theorem	wl-clabtv 37598*	Using class abstraction in a context, requiring 𝑥 and 𝜑 disjoint, but based on fewer axioms than wl-clabt 37599. (Contributed by Wolf Lammen, 29-May-2023.)
⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)})
Theorem	wl-clabt 37599	Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv 37598. (Contributed by Wolf Lammen, 29-May-2023.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)})
21.23  Mathbox for Brendan Leahy
Theorem	rabiun 37600*	Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.)
⊢ {𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∣ 𝜑} = ∪ 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝜑}