 Home Intuitionistic Logic ExplorerTheorem List (p. 21 of 95) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcesare 2001 "Cesare", one of the syllogisms of Aristotelian logic. No φ is ψ, and all χ is ψ, therefore no χ is φ. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 1996. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
x(φ → ¬ ψ)    &   x(χψ)       x(χ → ¬ φ)

Theoremcamestres 2002 "Camestres", one of the syllogisms of Aristotelian logic. All φ is ψ, and no χ is ψ, therefore no χ is φ. (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.)
x(φψ)    &   x(χ → ¬ ψ)       x(χ → ¬ φ)

Theoremfestino 2003 "Festino", one of the syllogisms of Aristotelian logic. No φ is ψ, and some χ is ψ, therefore some χ is not φ. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
x(φ → ¬ ψ)    &   x(χ ψ)       x(χ ¬ φ)

Theorembaroco 2004 "Baroco", one of the syllogisms of Aristotelian logic. All φ is ψ, and some χ is not ψ, therefore some χ is not φ. (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.)
x(φψ)    &   x(χ ¬ ψ)       x(χ ¬ φ)

Theoremcesaro 2005 "Cesaro", one of the syllogisms of Aristotelian logic. No φ is ψ, all χ is ψ, and χ exist, therefore some χ is not φ. (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.)
x(φ → ¬ ψ)    &   x(χψ)    &   xχ       x(χ ¬ φ)

Theoremcamestros 2006 "Camestros", one of the syllogisms of Aristotelian logic. All φ is ψ, no χ is ψ, and χ exist, therefore some χ is not φ. (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.)
x(φψ)    &   x(χ → ¬ ψ)    &   xχ       x(χ ¬ φ)

Theoremdatisi 2007 "Datisi", one of the syllogisms of Aristotelian logic. All φ is ψ, and some φ is χ, therefore some χ is ψ. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
x(φψ)    &   x(φ χ)       x(χ ψ)

Theoremdisamis 2008 "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.)
x(φ ψ)    &   x(φχ)       x(χ ψ)

Theoremferison 2009 "Ferison", one of the syllogisms of Aristotelian logic. No φ is ψ, and some φ is χ, therefore some χ is not ψ. (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.)
x(φ → ¬ ψ)    &   x(φ χ)       x(χ ¬ ψ)

Theorembocardo 2010 "Bocardo", one of the syllogisms of Aristotelian logic. Some φ is not ψ, and all φ is χ, therefore some χ is not ψ. (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 2008; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
x(φ ¬ ψ)    &   x(φχ)       x(χ ¬ ψ)

Theoremfelapton 2011 "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. (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.)
x(φ → ¬ ψ)    &   x(φχ)    &   xφ       x(χ ¬ ψ)

Theoremdarapti 2012 "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.)
x(φψ)    &   x(φχ)    &   xφ       x(χ ψ)

Theoremcalemes 2013 "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.) (Revised by David A. Wheeler, 2-Sep-2016.)
x(φψ)    &   x(ψ → ¬ χ)       x(χ → ¬ φ)

Theoremdimatis 2014 "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 1997 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
x(φ ψ)    &   x(ψχ)       x(χ φ)

Theoremfresison 2015 "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.) (Revised by David A. Wheeler, 2-Sep-2016.)
x(φ → ¬ ψ)    &   x(ψ χ)       x(χ ¬ φ)

Theoremcalemos 2016 "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.) (Revised by David A. Wheeler, 2-Sep-2016.)
x(φψ)    &   x(ψ → ¬ χ)    &   xχ       x(χ ¬ φ)

Theoremfesapo 2017 "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.) (Revised by David A. Wheeler, 2-Sep-2016.)
x(φ → ¬ ψ)    &   x(ψχ)    &   xψ       x(χ ¬ φ)

Theorembamalip 2018 "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.) Like barbari 1999. (Contributed by David A. Wheeler, 28-Aug-2016.)
x(φψ)    &   x(ψχ)    &   xφ       x(χ φ)

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 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 Axiom of Restricted Separation would require us to develop the ability to classify formulas as bounded formulas, similar to the machinery we have built up for asserting on whether variables are free in formulas.

2.1.1  Introduce the Axiom of Extensionality

Axiomax-ext 2019* 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 on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes (w(w xw y) → (x zy z)), and equality x = y is defined as w(w xw 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 1392 through ax-16 1692 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 2019 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 2019 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.)

(z(z xz y) → x = y)

Theoremaxext3 2020* 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.)
(z(z xz y) → x = y)

Theoremaxext4 2021* 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 2019. (Contributed by NM, 14-Nov-2008.)
(x = yz(z xz y))

Theorembm1.1 2022* 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.)
xφ       (xy(y xφ) → ∃!xy(y xφ))

2.1.2  Class abstractions (a.k.a. class builders)

Syntaxcab 2023 Introduce the class builder or class abstraction notation ("the class of sets x such that φ 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 2032, justifying the assignment of setvar variables to class variables via the use of cv 1241.
class {xφ}

Definitiondf-clab 2024 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, φ will have y as a free variable, and "{yφ} " is read "the class of all sets y such that φ(y) is true." We do not define {yφ} in isolation but only as part of an expression that extends or "overloads" the relationship.

This is our first use of the symbol to connect classes instead of sets. The syntax definition wcel 1390, which extends or "overloads" the wel 1391 definition connecting setvar variables, requires that both sides of be a class. In df-cleq 2030 and df-clel 2033, we introduce a new kind of variable (class variable) that can substituted with expressions such as {yφ}. In the present definition, the x on the left-hand side is a setvar variable. Syntax definition cv 1241 allows us to substitute a setvar variable x for a class variable: all sets are classes by cvjust 2032 (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 2143 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 {yφ} a "class term".

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

(x {yφ} ↔ [x / y]φ)

Theoremabid 2025 Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.)
(x {xφ} ↔ φ)

Theoremhbab1 2026* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)
(y {xφ} → x y {xφ})

Theoremnfsab1 2027* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
x y {xφ}

Theoremhbab 2028* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
(φxφ)       (z {yφ} → x z {yφ})

Theoremnfsab 2029* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       x z {yφ}

Definitiondf-cleq 2030* 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 = zx(x yx z), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2021). 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 yx = 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.

In the form of dfcleq 2031, 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 http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 15-Sep-1993.)

(x(x yx z) → y = z)       (A = Bx(x Ax B))

Theoremdfcleq 2031* The same as df-cleq 2030 with the hypothesis removed using the Axiom of Extensionality ax-ext 2019. (Contributed by NM, 15-Sep-1993.)
(A = Bx(x Ax B))

Theoremcvjust 2032* 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 1241, which allows us to substitute a setvar variable for a class variable. See also cab 2023 and df-clab 2024. Note that this is not a rigorous justification, because cv 1241 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 = {yy x}

Definitiondf-clel 2033* 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 2030 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2030 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1810), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2024.

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 http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 5-Aug-1993.)

(A Bx(x = A x B))

Theoremeqriv 2034* Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(x Ax B)       A = B

Theoremeqrdv 2035* Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.)
(φ → (x Ax B))       (φA = B)

Theoremeqrdav 2036* Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.)
((φ x A) → x 𝐶)    &   ((φ x B) → x 𝐶)    &   ((φ x 𝐶) → (x Ax B))       (φA = B)

Theoremeqid 2037 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 2038 Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
(φA = A)

Theoremeqcom 2039 Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.)
(A = BB = A)

Theoremeqcoms 2040 Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.)
(A = Bφ)       (B = Aφ)

Theoremeqcomi 2041 Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.)
A = B       B = A

Theoremeqcomd 2042 Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.)
(φA = B)       (φB = A)

Theoremeqeq1 2043 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
(A = B → (A = 𝐶B = 𝐶))

Theoremeqeq1i 2044 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
A = B       (A = 𝐶B = 𝐶)

Theoremeqeq1d 2045 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
(φA = B)       (φ → (A = 𝐶B = 𝐶))

Theoremeqeq2 2046 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
(A = B → (𝐶 = A𝐶 = B))

Theoremeqeq2i 2047 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
A = B       (𝐶 = A𝐶 = B)

Theoremeqeq2d 2048 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
(φA = B)       (φ → (𝐶 = A𝐶 = B))

Theoremeqeq12 2049 Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
((A = B 𝐶 = 𝐷) → (A = 𝐶B = 𝐷))

Theoremeqeq12i 2050 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 = 𝐶B = 𝐷)

Theoremeqeq12d 2051 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A = 𝐶B = 𝐷))

Theoremeqeqan12d 2052 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φA = B)    &   (ψ𝐶 = 𝐷)       ((φ ψ) → (A = 𝐶B = 𝐷))

Theoremeqeqan12rd 2053 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
(φA = B)    &   (ψ𝐶 = 𝐷)       ((ψ φ) → (A = 𝐶B = 𝐷))

Theoremeqtr 2054 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 2055 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 2056 A transitive law for class equality. (Contributed by NM, 20-May-2005.)
((A = 𝐶 B = 𝐶) → A = B)

Theoremeqtri 2057 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
A = B    &   B = 𝐶       A = 𝐶

Theoremeqtr2i 2058 An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
A = B    &   B = 𝐶       𝐶 = A

Theoremeqtr3i 2059 An equality transitivity inference. (Contributed by NM, 6-May-1994.)
A = B    &   A = 𝐶       B = 𝐶

Theoremeqtr4i 2060 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
A = B    &   𝐶 = B       A = 𝐶

Theorem3eqtri 2061 An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
A = B    &   B = 𝐶    &   𝐶 = 𝐷       A = 𝐷

Theorem3eqtrri 2062 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 2063 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
A = B    &   𝐶 = B    &   𝐶 = 𝐷       A = 𝐷

Theorem3eqtr2ri 2064 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 2065 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 2066 An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
A = B    &   A = 𝐶    &   B = 𝐷       𝐷 = 𝐶

Theorem3eqtr4i 2067 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 2068 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 2069 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(φA = B)    &   (φB = 𝐶)       (φA = 𝐶)

Theoremeqtr2d 2070 An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
(φA = B)    &   (φB = 𝐶)       (φ𝐶 = A)

Theoremeqtr3d 2071 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
(φA = B)    &   (φA = 𝐶)       (φB = 𝐶)

Theoremeqtr4d 2072 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
(φA = B)    &   (φ𝐶 = B)       (φA = 𝐶)

Theorem3eqtrd 2073 A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
(φA = B)    &   (φB = 𝐶)    &   (φ𝐶 = 𝐷)       (φA = 𝐷)

Theorem3eqtrrd 2074 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φA = B)    &   (φB = 𝐶)    &   (φ𝐶 = 𝐷)       (φ𝐷 = A)

Theorem3eqtr2d 2075 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
(φA = B)    &   (φ𝐶 = B)    &   (φ𝐶 = 𝐷)       (φA = 𝐷)

Theorem3eqtr2rd 2076 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
(φA = B)    &   (φ𝐶 = B)    &   (φ𝐶 = 𝐷)       (φ𝐷 = A)

Theorem3eqtr3d 2077 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φA = B)    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ𝐶 = 𝐷)

Theorem3eqtr3rd 2078 A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
(φA = B)    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ𝐷 = 𝐶)

Theorem3eqtr4d 2079 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φA = B)    &   (φ𝐶 = A)    &   (φ𝐷 = B)       (φ𝐶 = 𝐷)

Theorem3eqtr4rd 2080 A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
(φA = B)    &   (φ𝐶 = A)    &   (φ𝐷 = B)       (φ𝐷 = 𝐶)

Theoremsyl5eq 2081 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
A = B    &   (φB = 𝐶)       (φA = 𝐶)

Theoremsyl5req 2082 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
A = B    &   (φB = 𝐶)       (φ𝐶 = A)

Theoremsyl5eqr 2083 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
B = A    &   (φB = 𝐶)       (φA = 𝐶)

Theoremsyl5reqr 2084 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
B = A    &   (φB = 𝐶)       (φ𝐶 = A)

Theoremsyl6eq 2085 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(φA = B)    &   B = 𝐶       (φA = 𝐶)

Theoremsyl6req 2086 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
(φA = B)    &   B = 𝐶       (φ𝐶 = A)

Theoremsyl6eqr 2087 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(φA = B)    &   𝐶 = B       (φA = 𝐶)

Theoremsyl6reqr 2088 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
(φA = B)    &   𝐶 = B       (φ𝐶 = A)

Theoremsylan9eq 2089 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φA = B)    &   (ψB = 𝐶)       ((φ ψ) → A = 𝐶)

Theoremsylan9req 2090 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
(φB = A)    &   (ψB = 𝐶)       ((φ ψ) → A = 𝐶)

Theoremsylan9eqr 2091 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
(φA = B)    &   (ψB = 𝐶)       ((ψ φ) → A = 𝐶)

Theorem3eqtr3g 2092 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
(φA = B)    &   A = 𝐶    &   B = 𝐷       (φ𝐶 = 𝐷)

Theorem3eqtr3a 2093 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
A = B    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ𝐶 = 𝐷)

Theorem3eqtr4g 2094 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
(φA = B)    &   𝐶 = A    &   𝐷 = B       (φ𝐶 = 𝐷)

Theorem3eqtr4a 2095 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    &   (φ𝐶 = A)    &   (φ𝐷 = B)       (φ𝐶 = 𝐷)

Theoremeq2tri 2096 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
(A = 𝐶𝐷 = 𝐹)    &   (B = 𝐷𝐶 = 𝐺)       ((A = 𝐶 B = 𝐹) ↔ (B = 𝐷 A = 𝐺))

Theoremeleq1 2097 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(A = B → (A 𝐶B 𝐶))

Theoremeleq2 2098 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(A = B → (𝐶 A𝐶 B))

Theoremeleq12 2099 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
((A = B 𝐶 = 𝐷) → (A 𝐶B 𝐷))

Theoremeleq1i 2100 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
A = B       (A 𝐶B 𝐶)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9457
 Copyright terms: Public domain < Previous  Next >