P. 28 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 2701-2800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	disamis 2701	"Disamis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. In Aristotelian notation, IAI-3: MiP and MaS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∃𝑥(𝜑 ∧ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	ferison 2702	"Ferison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. Instance of datisi 2700. In Aristotelian notation, EIO-3: MeP and MiS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜑 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	bocardo 2703	"Bocardo", one of the syllogisms of Aristotelian logic. Some 𝜑 is not 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. Instance of disamis 2701. In Aristotelian notation, OAO-3: MoP and MaS therefore SoP. For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∃𝑥(𝜑 ∧ ¬ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	darapti 2704	"Darapti", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is 𝜓. In Aristotelian notation, AAI-3: MaP and MaS therefore SiP. For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	daraptiALT 2705	Alternate proof of darapti 2704, shorter but using more axioms. See comment of dariiALT 2686. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	felapton 2706	"Felapton", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is not 𝜓. Instance of darapti 2704. In Aristotelian notation, EAO-3: MeP and MaS therefore SoP. For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	calemes 2707	"Calemes", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜓 is 𝜒, therefore no 𝜒 is 𝜑. In Aristotelian notation, AEE-4: PaM and MeS therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → ¬ 𝜒)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	dimatis 2708	"Dimatis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜓 is 𝜒, therefore some 𝜒 is 𝜑. In Aristotelian notation, IAI-4: PiM and MaS therefore SiP. For example, "Some pets are rabbits", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2685 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∃𝑥(𝜑 ∧ 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜑)
Theorem	fresison 2709	"Fresison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓 (PeM), and some 𝜓 is 𝜒 (MiS), therefore some 𝜒 is not 𝜑 (SoP). In Aristotelian notation, EIO-4: PeM and MiS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜓 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	calemos 2710	"Calemos", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓 (PaM), no 𝜓 is 𝜒 (MeS), and 𝜒 exist, therefore some 𝜒 is not 𝜑 (SoP). In Aristotelian notation, AEO-4: PaM and MeS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → ¬ 𝜒)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	fesapo 2711	"Fesapo", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜓 exist, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EAO-4: PeM and MaS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    &   ⊢ ∃𝑥𝜓    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	bamalip 2712	"Bamalip", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜑 exist, therefore some 𝜒 is 𝜑. In Aristotelian notation, AAI-4: PaM and MaS therefore SiP. Very similar to barbari 2689. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜑)
1.7.2  Intuitionistic logic Intuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 8 and theorems such as exmid 903 or peirce 204. We mostly treat intuitionistic logic in a separate file, iset.mm, which is known as the Intuitionistic Logic Explorer on the web site. However, iset.mm has a number of additional axioms (mainly to replace definitions like df-or 857 and df-ex 1794 which are not valid in intuitionistic logic) and we want to prove those axioms here to demonstrate that adding those axioms in iset.mm does not make iset.mm any less consistent than set.mm. The following axioms are unchanged between set.mm and iset.mm: ax-1 6, ax-2 7, ax-mp 5, ax-4 1823, ax-11 2185, ax-gen 1809, ax-7 2022, ax-12 2206, ax-8 2138, ax-9 2146, and ax-5 1924. In this list of axioms, the ones that repeat earlier theorems are marked "(New usage is discouraged.)" so that the earlier theorems will be used consistently in other proofs.
Theorem	axia1 2713	Left 'and' elimination (intuitionistic logic axiom ax-ia1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜑)
Theorem	axia2 2714	Right 'and' elimination (intuitionistic logic axiom ax-ia2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜓)
Theorem	axia3 2715	'And' introduction (intuitionistic logic axiom ax-ia3). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
Theorem	axin1 2716	'Not' introduction (intuitionistic logic axiom ax-in1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
Theorem	axin2 2717	'Not' elimination (intuitionistic logic axiom ax-in2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ (¬ 𝜑 → (𝜑 → 𝜓))
Theorem	axio 2718	Definition of 'or' (intuitionistic logic axiom ax-io). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))
Theorem	axi4 2719	Specialization (intuitionistic logic axiom ax-4). This is just sp 2212 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	axi5r 2720	Converse of axc4 2347 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓))
Theorem	axial 2721	The setvar 𝑥 is not free in ∀𝑥𝜑 (intuitionistic logic axiom ax-ial). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	axie1 2722	The setvar 𝑥 is not free in ∃𝑥𝜑 (intuitionistic logic axiom ax-ie1). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	axie2 2723	A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.)
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	axi9 2724	Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-6 1981 but in intuitionistic logic it needs to be stated using the existential quantifier. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	axi10 2725	Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just axc11n 2451 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	axi12 2726	Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2407 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger. Usage of this theorem is discouraged because it depends on ax-13 2397. (Contributed by Jim Kingdon, 31-Dec-2017.) Avoid ax-11 2185. (Revised by Wolf Lammen, 24-Apr-2023.) (New usage is discouraged.)
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Theorem	axbnd 2727	Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2726 are fairly straightforward consequences of axc9 2407. But in intuitionistic logic, it is not easy to add the extra ∀𝑥 to axi12 2726 and so we treat the two as separate axioms. Usage of this theorem is discouraged because it depends on ax-13 2397. (Contributed by Jim Kingdon, 22-Mar-2018.) (Proof shortened by Wolf Lammen, 24-Apr-2023.) (New usage is discouraged.)
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets". A set can be an element of another set, and this relationship is indicated by the ∈ symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. A simplistic concept of sets, sometimes called "naive set theory", is vulnerable to a paradox called "Russell's Paradox" (ru 3737), a discovery that revolutionized the foundations of mathematics and logic. Russell's Paradox spawned the development of set theories that countered the paradox, including the ZF set theory that is most widely used and is defined here. Except for Extensionality, the axioms basically say, "given an arbitrary set x (and, in the cases of Replacement and Regularity, provided that an antecedent is satisfied), there exists another set y based on or constructed from it, with the stated properties". (The axiom of extensionality can also be restated this way as shown by axexte 2729.) The individual axiom links provide more detailed descriptions. We derive the redundant ZF axioms of Separation, Null Set, and Pairing from the others as theorems.
2.1  ZF Set Theory - start with the Axiom of Extensionality
2.1.1  Introduce the Axiom of Extensionality
Axiom	ax-ext 2728*	Axiom of extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461. Its converse is a theorem of predicate logic, elequ2g 2152. Set theory can also be formulated with a single primitive predicate ∈ on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes (∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)), and equality 𝑥 = 𝑦 is defined as ∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦). All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8. To use the above "equality-free" version of Extensionality with Metamath's predicate calculus axioms, we would rewrite all axioms involving equality with equality expanded according to the above definition. Some of those axioms may be provable from ax-ext and would become redundant, but this hasn't been studied carefully. General remarks: Our set theory axioms are presented using defined connectives (↔, ∃, etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives →, ¬, ∀, =, and ∈. It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so. It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable 𝑥 in ax-ext 2728 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both 𝑥 and 𝑧. This is in contrast to typical textbook presentations that present actual axioms (except for Replacement ax-rep 5221, which involves a wff metavariable). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the infinite axioms generated by the ax-ext 2728 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 21-May-1993.)
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	axexte 2729*	The axiom of extensionality (ax-ext 2728) restated so that it postulates the existence of a set 𝑧 given two arbitrary sets 𝑥 and 𝑦. This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.)
⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	axextg 2730*	A generalization of the axiom of extensionality in which 𝑥 and 𝑦 need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Remove dependencies on ax-10 2169, ax-12 2206, ax-13 2397. (Revised by BJ, 12-Jul-2019.) (Revised by Wolf Lammen, 9-Dec-2019.)
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	axextb 2731*	A bidirectional version of the axiom of extensionality. Although this theorem looks like a definition of equality, it requires the axiom of extensionality for its proof under our axiomatization. See the comments for ax-ext 2728 and df-cleq 2748. (Contributed by NM, 14-Nov-2008.)
⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
Theorem	axextmo 2732*	There exists at most one set with prescribed elements. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)
Theorem	nulmo 2733*	There exists at most one empty set. With either axnul 5249 or axnulALT 5248 or ax-nul 5250, this proves that there exists a unique empty set. In practice, once the language of classes is available, we use the stronger characterization among classes eq0 4297. (Contributed by NM, 22-Dec-2007.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.) (Proof shortened by Wolf Lammen, 26-Apr-2023.)
⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
2.1.2  Classes
2.1.2.1  Class abstractions
Syntax	cab 2734	Introduce the class abstraction (or class builder) notation: {𝑥 ∣ 𝜑} is the class of sets 𝑥 such that 𝜑(𝑥) is true. A setvar variable can be expressed as a class abstraction per Theorem cvjust 2750, justifying the substitution of class variables for setvar variables via the use of cv 1553.
class {𝑥 ∣ 𝜑}
Definition	df-clab 2735	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 2735 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 2748). Instead, df-clab 2735 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 2735 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 2728. Therefore, df-clab 2735 can be considered a definition only in systems that can prove ax-ext 2728 (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 2738). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑. 3. Remark 1 stresses that df-clab 2735 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 2138). 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 2136 and cab 2734, and it can be called an "extension" of the membership predicate because of wel 2137, whose proof uses cv 1553. An a posteriori justification for cv 1553 is given by cvjust 2750, 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 3737). 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 3516 which is used, for example, to convert elirrv 9535 to elirr 9538. 5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2735, df-cleq 2748, and df-clel 2831, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 30541), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2748 and df-clel 2831). 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 2728, see Appendix of [Levy] p. 357. 6. 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 2895 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2895. (Contributed by NM, 26-May-1993.) (Revised by BJ, 19-Aug-2023.)
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
Theorem	eleq1ab 2736	Extension (in the sense of Remark 3 of the comment of df-clab 2735) of elequ1 2143 from formulas of the form "setvar ∈ setvar" to formulas of the form "setvar ∈ class abstraction". This extension does not require ax-8 2138 contrary to elequ1 2143, but recall from Remark 3 of the comment of df-clab 2735 that it can be considered an extension only because of cvjust 2750, which does require ax-8 2138. This is an instance of eleq1w 2839 where the containing class is a class abstraction, and contrary to it, it can be proved without df-clel 2831. See also eleq1 2844 for general classes. The straightforward yet important fact that this statement can be proved from FOL= plus df-clab 2735 (hence without ax-ext 2728, df-cleq 2748 or df-clel 2831) was stressed by Mario Carneiro. (Contributed by BJ, 17-Aug-2023.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ 𝑦 ∈ {𝑧 ∣ 𝜑}))
Theorem	cleljustab 2737*	Extension of cleljust 2145 from formulas of the form "setvar ∈ setvar" to formulas of the form "setvar ∈ class abstraction". This is an instance of dfclel 2832 where the containing class is a class abstraction. The same remarks as for eleq1ab 2736 apply. (Contributed by BJ, 8-Nov-2021.) (Proof shortened by Steven Nguyen, 19-May-2023.)
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))
Theorem	abid 2738	Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 26-May-1993.)
⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
Theorem	vexwt 2739	A standard theorem of predicate calculus (stdpc4 2092) expressed using class abstractions. Closed form of vexw 2740. (Contributed by BJ, 14-Jun-2019.)
⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})
Theorem	vexw 2740	If 𝜑 is a theorem, then any set belongs to the class {𝑥 ∣ 𝜑}. Therefore, {𝑥 ∣ 𝜑} is "a" universal class. This is the closest one can get to defining a universal class, or proving vex 3452, without using ax-ext 2728. Note that this theorem has no disjoint variable condition and does not use df-clel 2831 nor df-cleq 2748 either: only propositional logic and ax-gen 1809 and df-clab 2735. This is sbt 2089 expressed using class abstractions. Without ax-ext 2728, one cannot define "the" universal class, since one could not prove for instance the justification theorem {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} (see vjust 3449). Indeed, in order to prove any equality of classes, one needs df-cleq 2748, which has ax-ext 2728 as a hypothesis. Therefore, the classes {𝑥 ∣ ⊤}, {𝑦 ∣ (𝜑 → 𝜑)}, {𝑧 ∣ (∀𝑡𝑡 = 𝑡 → ∀𝑡𝑡 = 𝑡)} and countless others are all universal classes whose equality cannot be proved without ax-ext 2728. Once dfcleq 2749 is available, we will define "the" universal class in df-v 3450. Its degenerate instance is also a simple consequence of abid 2738 (using mpbir 233). (Contributed by BJ, 13-Jun-2019.) Reduce axiom dependencies. (Revised by Steven Nguyen, 25-Apr-2023.)
⊢ 𝜑    ⇒   ⊢ 𝑦 ∈ {𝑥 ∣ 𝜑}
Theorem	vextru 2741	Every setvar is a member of {𝑥 ∣ ⊤}, which is therefore "a" universal class. Once class extensionality dfcleq 2749 is available, we can say "the" universal class (see df-v 3450). This is sbtru 2090 expressed using class abstractions. (Contributed by BJ, 2-Sep-2023.)
⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
Theorem	nfsab1 2742*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2206. (Revised by SN, 20-Sep-2023.)
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
Theorem	hbab1 2743*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2024.)
⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
Theorem	hbab 2744*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) Add disjoint variable condition to avoid ax-13 2397. See hbabg 2745 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
Theorem	hbabg 2745*	Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2397. See hbab 2744 for a version with more disjoint variable conditions, but not requiring ax-13 2397. (Contributed by NM, 1-Mar-1995.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
Theorem	nfsab 2746*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2397. See nfsabg 2747 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
Theorem	nfsabg 2747*	Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2397. See nfsab 2746 for a version with more disjoint variable conditions, but not requiring ax-13 2397. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
2.1.2.2  Class equality This section introduces class equality in df-cleq 2748. Note that apart from the local introduction of class variables to state the syntax axioms wceq 1554 and wcel 2136, this section is the first to use class variables. Therefore, the file set.mm contains declarations of class variables at the beginning of this section (not visible on the webpages).
Definition	df-cleq 2748*	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 2748 an axiom. See also comments under df-clab 2735, df-clel 2831, and eqabb 2895. In the form of dfcleq 2749, 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. While the three class definitions df-clab 2735, df-cleq 2748, and df-clel 2831 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 2831. (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	dfcleq 2749*	The defining characterization of class equality. It is proved, over Tarski's FOL, from the axiom of (set) extensionality (ax-ext 2728) and the definition of class equality (df-cleq 2748). Its forward implication is called "class extensionality". Remark: the proof uses axextb 2731 to prove also the hypothesis of df-cleq 2748 that is a degenerate instance, but it could be proved also from minimal propositional calculus and { ax-gen 1809, equid 2026 }. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	cvjust 2750*	Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1553, which allows to substitute a setvar variable for a class variable. See also cab 2734 and df-clab 2735. Note that this is not a rigorous justification, because cv 1553 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." See abid1 2892 for the version of cvjust 2750 extended to classes. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2397. (Revised by Wolf Lammen, 4-May-2023.)
⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥}
Theorem	ax9ALT 2751	Proof of ax-9 2146 from Tarski's FOL and dfcleq 2749. For a version not using ax-8 2138 either, see eleq2w2 2752. This shows that dfcleq 2749 is too powerful to be used as a definition instead of df-cleq 2748. Note that ax-ext 2728 is also a direct consequence of dfcleq 2749 (as an instance of its forward implication). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
Theorem	eleq2w2 2752*	A weaker version of eleq2 2845 (but stronger than ax-9 2146 and elequ2 2151) that uses ax-12 2206 to avoid ax-8 2138 and df-clel 2831. Compare eleq2w 2840, whose setvars appear where the class variables are in this theorem, and vice versa. (Contributed by BJ, 24-Jun-2019.) Strengthen from setvar variables to class variables. (Revised by WL and SN, 23-Aug-2024.)
⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	eqriv 2753*	Infer equality of classes from equivalence of membership. (Contributed by NM, 21-Jun-1993.)
⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqrdv 2754*	Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqrdav 2755*	Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqid 2756	Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41. This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also biid 263. An early mention of this law can be found in Aristotle, Metaphysics, Z.17, 1041a10-20. (Thanks to Stefan Allan and BJ for this information.) (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Oct-2017.)
⊢ 𝐴 = 𝐴
Theorem	eqidd 2757	Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
⊢ (𝜑 → 𝐴 = 𝐴)
Theorem	eqeq1d 2758	Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
Theorem	eqeq1dALT 2759	Alternate proof of eqeq1d 2758, shorter but requiring ax-12 2206. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 19-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
Theorem	eqeq1 2760	Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
Theorem	eqeq1i 2761	Inference from equality to equivalence of equalities. (Contributed by NM, 15-Jul-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)
Theorem	eqcomd 2762	Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) Allow shortening of eqcom 2763. (Revised by Wolf Lammen, 19-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐴)
Theorem	eqcom 2763	Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
Theorem	eqcoms 2764	Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 24-Jun-1993.)
⊢ (𝐴 = 𝐵 → 𝜑)    ⇒   ⊢ (𝐵 = 𝐴 → 𝜑)
Theorem	eqcomi 2765	Inference from commutative law for class equality. (Contributed by NM, 26-May-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ 𝐵 = 𝐴
Theorem	neqcomd 2766	Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐵 = 𝐴)
Theorem	eqeq2d 2767	Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Allow shortening of eqeq2 2768. (Revised by Wolf Lammen, 19-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵))
Theorem	eqeq2 2768	Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵))
Theorem	eqeq2i 2769	Inference from equality to equivalence of equalities. (Contributed by NM, 26-May-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)
Theorem	eqeqan12d 2770	A useful inference for substituting definitions into an equality. See also eqeqan12dALT 2775. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) Shorten other proofs. (Revised by Wolf Lammen, 23-Oct-2024.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	eqeqan12rd 2771	A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	eqeq12d 2772	A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Oct-2024.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	eqeq12 2773	Equality relationship among four classes. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	eqeq12i 2774	A useful inference for substituting definitions into an equality. (Contributed by NM, 15-Jul-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)
Theorem	eqeqan12dALT 2775	Alternate proof of eqeqan12d 2770. This proof has one more step but one fewer essential step. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	eqtr 2776	Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶)
Theorem	eqtr2 2777	A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Oct-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)
Theorem	eqtr3 2778	A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Wolf Lammen, 24-Oct-2024.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
Theorem	eqtri 2779	An equality transitivity inference. (Contributed by NM, 26-May-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 = 𝐶
Theorem	eqtr2i 2780	An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐶 = 𝐴
Theorem	eqtr3i 2781	An equality transitivity inference. (Contributed by NM, 6-May-1994.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 = 𝐶    ⇒   ⊢ 𝐵 = 𝐶
Theorem	eqtr4i 2782	An equality transitivity inference. (Contributed by NM, 26-May-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 = 𝐶
Theorem	3eqtri 2783	An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐴 = 𝐷
Theorem	3eqtrri 2784	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐷 = 𝐴
Theorem	3eqtr2i 2785	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐴 = 𝐷
Theorem	3eqtr2ri 2786	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐷 = 𝐴
Theorem	3eqtr3i 2787	An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐶 = 𝐷
Theorem	3eqtr3ri 2788	An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐷 = 𝐶
Theorem	3eqtr4i 2789	An inference from three chained equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐶 = 𝐷
Theorem	3eqtr4ri 2790	An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐷 = 𝐶
Theorem	eqtrd 2791	An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	eqtr2d 2792	An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	eqtr3d 2793	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	eqtr4d 2794	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	3eqtrd 2795	A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐷)
Theorem	3eqtrrd 2796	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐴)
Theorem	3eqtr2d 2797	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐷)
Theorem	3eqtr2rd 2798	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐴)
Theorem	3eqtr3d 2799	A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr3rd 2800	A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐶)