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Theorem List for Intuitionistic Logic Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembarbari 2101 "Barbari", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ch is  ph, and some  ch exist, therefore some  ch is  ps. (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  ph )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  ps )
 
Theoremcelaront 2102 "Celaront", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ch is  ph, and some  ch exist, therefore some  ch is not  ps. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ph )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ps )
 
Theoremcesare 2103 "Cesare", one of the syllogisms of Aristotelian logic. No  ph is  ps, and all  ch is  ps, therefore no  ch is  ph. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2098. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ps )   =>    |-  A. x ( ch  ->  -.  ph )
 
Theoremcamestres 2104 "Camestres", one of the syllogisms of Aristotelian logic. All  ph is  ps, and no  ch is  ps, therefore no  ch is  ph. (In Aristotelian notation, AEE-2: PaM and SeM therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  -.  ps )   =>    |-  A. x ( ch  ->  -.  ph )
 
Theoremfestino 2105 "Festino", one of the syllogisms of Aristotelian logic. No  ph is  ps, and some  ch is  ps, therefore some  ch is not  ph. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ch  /\  ps )   =>    |-  E. x ( ch  /\  -.  ph )
 
Theorembaroco 2106 "Baroco", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ch is not  ps, therefore some  ch is not  ph. (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  E. x ( ch  /\  -.  ps )   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremcesaro 2107 "Cesaro", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ch is  ps, and  ch exist, therefore some  ch is not  ph. (In Aristotelian notation, EAO-2: PeM and SaM therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ps )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremcamestros 2108 "Camestros", one of the syllogisms of Aristotelian logic. All  ph is  ps, no  ch is  ps, and  ch exist, therefore some  ch is not  ph. (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  -.  ps )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremdatisi 2109 "Datisi", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ph is  ch, therefore some  ch is  ps. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  E. x (
 ph  /\  ch )   =>    |-  E. x ( ch  /\  ps )
 
Theoremdisamis 2110 "Disamis", one of the syllogisms of Aristotelian logic. Some  ph is  ps, and all  ph is  ch, therefore some  ch is  ps. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 E. x ( ph  /\ 
 ps )   &    |-  A. x (
 ph  ->  ch )   =>    |- 
 E. x ( ch 
 /\  ps )
 
Theoremferison 2111 "Ferison", one of the syllogisms of Aristotelian logic. No  ph is  ps, and some  ph is  ch, therefore some  ch is not  ps. (In Aristotelian notation, EIO-3: MeP and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ph  /\  ch )   =>    |-  E. x ( ch  /\  -.  ps )
 
Theorembocardo 2112 "Bocardo", one of the syllogisms of Aristotelian logic. Some  ph is not  ps, and all  ph is  ch, therefore some  ch is not  ps. (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". A reorder of disamis 2110; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
 |- 
 E. x ( ph  /\ 
 -.  ps )   &    |-  A. x (
 ph  ->  ch )   =>    |- 
 E. x ( ch 
 /\  -.  ps )
 
Theoremfelapton 2113 "Felapton", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ph is  ch, and some  ph exist, therefore some  ch is not  ps. (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.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ph  ->  ch )   &    |-  E. x ph   =>    |-  E. x ( ch  /\  -. 
 ps )
 
Theoremdarapti 2114 "Darapti", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ph is  ch, and some  ph exist, therefore some  ch is  ps. (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.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x (
 ph  ->  ch )   &    |-  E. x ph   =>    |-  E. x ( ch  /\  ps )
 
Theoremcalemes 2115 "Calemes", one of the syllogisms of Aristotelian logic. All  ph is  ps, and no  ps is  ch, therefore no  ch is  ph. (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ps  ->  -.  ch )   =>    |-  A. x ( ch  ->  -.  ph )
 
Theoremdimatis 2116 "Dimatis", one of the syllogisms of Aristotelian logic. Some  ph is  ps, and all  ps is  ch, therefore some  ch is  ph. (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 2099 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 E. x ( ph  /\ 
 ps )   &    |-  A. x ( ps  ->  ch )   =>    |-  E. x ( ch  /\  ph )
 
Theoremfresison 2117 "Fresison", one of the syllogisms of Aristotelian logic. No  ph is  ps (PeM), and some  ps is  ch (MiS), therefore some  ch is not  ph (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ps  /\  ch )   =>    |-  E. x ( ch  /\  -.  ph )
 
Theoremcalemos 2118 "Calemos", one of the syllogisms of Aristotelian logic. All  ph is  ps (PaM), no  ps is  ch (MeS), and  ch exist, therefore some  ch is not  ph (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ps  ->  -.  ch )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremfesapo 2119 "Fesapo", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ps is  ch, and  ps exist, therefore some  ch is not  ph. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ps  ->  ch )   &    |-  E. x ps   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theorembamalip 2120 "Bamalip", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ps is  ch, and  ph exist, therefore some  ch is  ph. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2101. (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ps  ->  ch )   &    |-  E. x ph   =>    |-  E. x ( ch  /\  ph )
 
PART 2  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  e. 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.

Here we develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in Metamath because the statement of some of its axioms uses the notion of "bounded formula". Since Metamath has, purposefully, a very weak metalogic, that notion must be developed in the logic itself. This is similar to our treatment of substitution (df-sb 1736) and our definition of the nonfreeness predicate (df-nf 1437), whereas substitution and bound and free variables are ordinarily defined in the metalogic. The development of CZF has begun in BJ's mathbox, see wbd 13069.

 
2.1  IZF Set Theory - start with the Axiom of Extensionality
 
2.1.1  Introduce the Axiom of Extensionality
 
Axiomax-ext 2121* Axiom of Extensionality. It states that two sets are identical if they contain the same elements. Axiom 1 of [Crosilla] p. "Axioms of CZF and IZF" (with unnecessary quantifiers removed).

Set theory can also be formulated with a single primitive predicate  e. on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes  ( A. w
( w  e.  x  <->  w  e.  y )  -> 
( x  e.  z  ->  y  e.  z ) ), and equality  x  =  y is defined as  A. w ( w  e.  x  <->  w  e.  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 logical axioms, we would rewrite ax-8 1482 through ax-16 1786 with equality expanded according to the above definition. Some of those axioms could be proved from set theory and would be redundant. Not all of them are redundant, since our axioms of predicate calculus make essential use of equality for the proper substitution that is a primitive notion in traditional predicate calculus. A study of such an axiomatization would be an interesting project for someone exploring the foundations of logic.

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 2121 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 axioms which involve wff metavariables). 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 2121 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.)

 |-  ( A. z ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y )
 
Theoremaxext3 2122* 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.)
 |-  ( A. z ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y )
 
Theoremaxext4 2123* A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2121. (Contributed by NM, 14-Nov-2008.)
 |-  ( x  =  y  <->  A. z ( z  e.  x  <->  z  e.  y
 ) )
 
Theorembm1.1 2124* Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.)
 |- 
 F/ x ph   =>    |-  ( E. x A. y ( y  e.  x  <->  ph )  ->  E! x A. y ( y  e.  x  <->  ph ) )
 
2.1.2  Class abstractions (a.k.a. class builders)
 
Syntaxcab 2125 Introduce the class builder or class abstraction notation ("the class of sets  x such that  ph is true"). Our class variables  A,  B, etc. range over class builders (sometimes implicitly). Note that a setvar variable can be expressed as a class builder per theorem cvjust 2134, justifying the assignment of setvar variables to class variables via the use of cv 1330.
 class  { x  |  ph }
 
Definitiondf-clab 2126 Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature.  x and  y need not be distinct. Definition 2.1 of [Quine] p. 16. Typically,  ph will have  y as a free variable, and " { y  |  ph } " is read "the class of all sets  y such that  ph ( y ) is true." We do not define  { y  |  ph } in isolation but only as part of an expression that extends or "overloads" the  e. relationship.

This is our first use of the 
e. symbol to connect classes instead of sets. The syntax definition wcel 1480, which extends or "overloads" the wel 1481 definition connecting setvar variables, requires that both sides of  e. be a class. In df-cleq 2132 and df-clel 2135, we introduce a new kind of variable (class variable) that can substituted with expressions such as  { y  | 
ph }. In the present definition, the  x on the left-hand side is a setvar variable. Syntax definition cv 1330 allows us to substitute a setvar variable  x for a class variable: all sets are classes by cvjust 2134 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 2248 for a quick overview).

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 called the "axiom of class comprehension" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. He calls the construction  {
y  |  ph } a "class term".

For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class 2248. (Contributed by NM, 5-Aug-1993.)

 |-  ( x  e.  {
 y  |  ph }  <->  [ x  /  y ] ph )
 
Theoremabid 2127 Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  { x  |  ph }  <->  ph )
 
Theoremhbab1 2128* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)
 |-  ( y  e.  { x  |  ph }  ->  A. x  y  e.  { x  |  ph } )
 
Theoremnfsab1 2129* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x  y  e. 
 { x  |  ph }
 
Theoremhbab 2130* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( z  e.  {
 y  |  ph }  ->  A. x  z  e.  {
 y  |  ph } )
 
Theoremnfsab 2131* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x  z  e. 
 { y  |  ph }
 
Definitiondf-cleq 2132* 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.

This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce  y  =  z  <->  A. x ( x  e.  y  <->  x  e.  z
), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2123). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated.

We could avoid this complication by introducing a new symbol, say =2, in place of  =. This would also have the advantage of making elimination of the definition straightforward, so that we could eliminate Extensionality as a hypothesis. We would then also have the advantage of being able to identify in various proofs exactly where Extensionality truly comes into play rather than just being an artifact of a definition. One of our theorems would then be  x =2  y  <->  x  =  y by invoking Extensionality.

However, to conform to literature usage, we retain this overloaded definition. This also makes some proofs shorter and probably easier to read, without the constant switching between two kinds of equality.

See also comments under df-clab 2126, df-clel 2135, and abeq2 2248.

In the form of dfcleq 2133, 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.

For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class 2133. (Contributed by NM, 15-Sep-1993.)

 |-  ( A. x ( x  e.  y  <->  x  e.  z
 )  ->  y  =  z )   =>    |-  ( A  =  B  <->  A. x ( x  e.  A  <->  x  e.  B ) )
 
Theoremdfcleq 2133* The same as df-cleq 2132 with the hypothesis removed using the Axiom of Extensionality ax-ext 2121. (Contributed by NM, 15-Sep-1993.)
 |-  ( A  =  B  <->  A. x ( x  e.  A  <->  x  e.  B ) )
 
Theoremcvjust 2134* 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 1330, which allows us to substitute a setvar variable for a class variable. See also cab 2125 and df-clab 2126. Note that this is not a rigorous justification, because cv 1330 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." (Contributed by NM, 7-Nov-2006.)
 |-  x  =  { y  |  y  e.  x }
 
Definitiondf-clel 2135* 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. Note that like df-cleq 2132 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2132 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1910), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2126.

This is called the "axiom of membership" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms.

For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class 2126. (Contributed by NM, 5-Aug-1993.)

 |-  ( A  e.  B  <->  E. x ( x  =  A  /\  x  e.  B ) )
 
Theoremeqriv 2136* Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  A  <->  x  e.  B )   =>    |-  A  =  B
 
Theoremeqrdv 2137* Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.)
 |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqrdav 2138* Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.)
 |-  ( ( ph  /\  x  e.  A )  ->  x  e.  C )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  x  e.  C )   &    |-  ( ( ph  /\  x  e.  C )  ->  ( x  e.  A  <->  x  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqid 2139 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Zeta, 17, 1041 a, 10-20). (Thanks to Stefan Allan and BJ for this information.) (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 14-Oct-2017.)

 |-  A  =  A
 
Theoremeqidd 2140 Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
 |-  ( ph  ->  A  =  A )
 
Theoremeqcom 2141 Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  <->  B  =  A )
 
Theoremeqcoms 2142 Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  -> 
 ph )   =>    |-  ( B  =  A  -> 
 ph )
 
Theoremeqcomi 2143 Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  B  =  A
 
Theoremneqcomd 2144 Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)
 |-  ( ph  ->  -.  A  =  B )   =>    |-  ( ph  ->  -.  B  =  A )
 
Theoremeqcomd 2145 Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  B  =  A )
 
Theoremeqeq1 2146 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
 
Theoremeqeq1i 2147 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  =  C 
 <->  B  =  C )
 
Theoremeqeq1d 2148 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  =  C  <->  B  =  C ) )
 
Theoremeqeq2 2149 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( C  =  A  <->  C  =  B ) )
 
Theoremeqeq2i 2150 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  =  A 
 <->  C  =  B )
 
Theoremeqeq2d 2151 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  =  A  <->  C  =  B ) )
 
Theoremeqeq12 2152 Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  =  C 
 <->  B  =  D ) )
 
Theoremeqeq12i 2153 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  =  C  <->  B  =  D )
 
Theoremeqeq12d 2154 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremeqeqan12d 2155 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremeqeqan12rd 2156 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ps 
 /\  ph )  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremeqtr 2157 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 )
 
Theoremeqtr2 2158 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( A  =  B  /\  A  =  C )  ->  B  =  C )
 
Theoremeqtr3 2159 A transitive law for class equality. (Contributed by NM, 20-May-2005.)
 |-  ( ( A  =  C  /\  B  =  C )  ->  A  =  B )
 
Theoremeqtri 2160 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B  =  C   =>    |-  A  =  C
 
Theoremeqtr2i 2161 An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
 |-  A  =  B   &    |-  B  =  C   =>    |-  C  =  A
 
Theoremeqtr3i 2162 An equality transitivity inference. (Contributed by NM, 6-May-1994.)
 |-  A  =  B   &    |-  A  =  C   =>    |-  B  =  C
 
Theoremeqtr4i 2163 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  C  =  B   =>    |-  A  =  C
 
Theorem3eqtri 2164 An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
 |-  A  =  B   &    |-  B  =  C   &    |-  C  =  D   =>    |-  A  =  D
 
Theorem3eqtrri 2165 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
 
Theorem3eqtr2i 2166 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
 |-  A  =  B   &    |-  C  =  B   &    |-  C  =  D   =>    |-  A  =  D
 
Theorem3eqtr2ri 2167 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
 
Theorem3eqtr3i 2168 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
 
Theorem3eqtr3ri 2169 An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
 |-  A  =  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  D  =  C
 
Theorem3eqtr4i 2170 An inference from three chained equalities. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  A  =  B   &    |-  C  =  A   &    |-  D  =  B   =>    |-  C  =  D
 
Theorem3eqtr4ri 2171 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
 
Theoremeqtrd 2172 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremeqtr2d 2173 An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  C  =  A )
 
Theoremeqtr3d 2174 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B  =  C )
 
Theoremeqtr4d 2175 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  =  C )
 
Theorem3eqtrd 2176 A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  A  =  D )
 
Theorem3eqtrrd 2177 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  D  =  A )
 
Theorem3eqtr2d 2178 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  A  =  D )
 
Theorem3eqtr2rd 2179 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  D  =  A )
 
Theorem3eqtr3d 2180 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr3rd 2181 A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  D  =  C )
 
Theorem3eqtr4d 2182 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr4rd 2183 A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  D  =  C )
 
Theoremsyl5eq 2184 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl5req 2185 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  A  =  B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  C  =  A )
 
Theoremsyl5eqr 2186 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  B  =  A   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl5reqr 2187 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  B  =  A   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  C  =  A )
 
Theoremsyl6eq 2188 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl6req 2189 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   &    |-  B  =  C   =>    |-  ( ph  ->  C  =  A )
 
Theoremsyl6eqr 2190 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  =  C )
 
Theoremsyl6reqr 2191 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   &    |-  C  =  B   =>    |-  ( ph  ->  C  =  A )
 
Theoremsylan9eq 2192 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  B  =  C )   =>    |-  ( ( ph  /\ 
 ps )  ->  A  =  C )
 
Theoremsylan9req 2193 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
 |-  ( ph  ->  B  =  A )   &    |-  ( ps  ->  B  =  C )   =>    |-  ( ( ph  /\ 
 ps )  ->  A  =  C )
 
Theoremsylan9eqr 2194 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  B  =  C )   =>    |-  ( ( ps 
 /\  ph )  ->  A  =  C )
 
Theorem3eqtr3g 2195 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
 |-  ( ph  ->  A  =  B )   &    |-  A  =  C   &    |-  B  =  D   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr3a 2196 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |-  A  =  B   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr4g 2197 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  C  =  A   &    |-  D  =  B   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr4a 2198 A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  A  =  B   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C  =  D )
 
Theoremeq2tri 2199 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
 |-  ( A  =  C  ->  D  =  F )   &    |-  ( B  =  D  ->  C  =  G )   =>    |-  ( ( A  =  C  /\  B  =  F ) 
 <->  ( B  =  D  /\  A  =  G ) )
 
Theoremeleq1w 2200 Weaker version of eleq1 2202 (but more general than elequ1 1690) not depending on ax-ext 2121 nor df-cleq 2132. (Contributed by BJ, 24-Jun-2019.)
 |-  ( x  =  y 
 ->  ( x  e.  A  <->  y  e.  A ) )
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