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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | darapti 2001 | "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(φ → ψ) & ⊢ ∀x(φ → χ) & ⊢ ∃xφ ⇒ ⊢ ∃x(χ ∧ ψ) | ||
Theorem | calemes 2002 | "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(φ → ψ) & ⊢ ∀x(ψ → ¬ χ) ⇒ ⊢ ∀x(χ → ¬ φ) | ||
Theorem | dimatis 2003 | "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 1986 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) |
⊢ ∃x(φ ∧ ψ) & ⊢ ∀x(ψ → χ) ⇒ ⊢ ∃x(χ ∧ φ) | ||
Theorem | fresison 2004 | "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) | ||
Theorem | calemos 2005 | "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(φ → ψ) & ⊢ ∀x(ψ → ¬ χ) & ⊢ ∃xχ ⇒ ⊢ ∃x(χ ∧ ¬ φ) | ||
Theorem | fesapo 2006 | "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(ψ → χ) & ⊢ ∃xψ ⇒ ⊢ ∃x(χ ∧ ¬ φ) | ||
Theorem | bamalip 2007 | "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 1988. (Contributed by David A. Wheeler, 28-Aug-2016.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(ψ → χ) & ⊢ ∃xφ ⇒ ⊢ ∃x(χ ∧ φ) | ||
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. | ||
Axiom | ax-ext 2008* |
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 ∈ 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 logical axioms, we would rewrite ax-8 1382 through ax-16 1683 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 2008 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 2008 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 ∈ x ↔ z ∈ y) → x = y) | ||
Theorem | axext3 2009* | 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 ∈ x ↔ z ∈ y) → x = y) | ||
Theorem | axext4 2010* | 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 2008. (Contributed by NM, 14-Nov-2008.) |
⊢ (x = y ↔ ∀z(z ∈ x ↔ z ∈ y)) | ||
Theorem | bm1.1 2011* | 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.) |
⊢ Ⅎxφ ⇒ ⊢ (∃x∀y(y ∈ x ↔ φ) → ∃!x∀y(y ∈ x ↔ φ)) | ||
Syntax | cab 2012 | 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 2021, justifying the assignment of setvar variables to class variables via the use of cv 1232. |
class {x ∣ φ} | ||
Definition | df-clab 2013 |
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 "{y ∣ φ} " is read "the class of
all sets y
such that φ(y) is true." We do not define {y ∣ φ} 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 1380, which extends or "overloads" the wel 1381 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2019 and df-clel 2022, we introduce a new kind of variable (class variable) that can substituted with expressions such as {y ∣ φ}. In the present definition, the x on the left-hand side is a setvar variable. Syntax definition cv 1232 allows us to substitute a setvar variable x for a class variable: all sets are classes by cvjust 2021 (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 2132 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 ∣ φ} 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 ∈ {y ∣ φ} ↔ [x / y]φ) | ||
Theorem | abid 2014 | Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.) |
⊢ (x ∈ {x ∣ φ} ↔ φ) | ||
Theorem | hbab1 2015* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.) |
⊢ (y ∈ {x ∣ φ} → ∀x y ∈ {x ∣ φ}) | ||
Theorem | nfsab1 2016* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎx y ∈ {x ∣ φ} | ||
Theorem | hbab 2017* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (z ∈ {y ∣ φ} → ∀x z ∈ {y ∣ φ}) | ||
Theorem | nfsab 2018* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎxφ ⇒ ⊢ Ⅎx z ∈ {y ∣ φ} | ||
Definition | df-cleq 2019* |
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 ↔ ∀x(x ∈ y ↔ x ∈ z), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2010). 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 2013, df-clel 2022, and abeq2 2132. In the form of dfcleq 2020, 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 ∈ y ↔ x ∈ z) → y = z) ⇒ ⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ B)) | ||
Theorem | dfcleq 2020* | The same as df-cleq 2019 with the hypothesis removed using the Axiom of Extensionality ax-ext 2008. (Contributed by NM, 15-Sep-1993.) |
⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ B)) | ||
Theorem | cvjust 2021* | 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 1232, which allows us to substitute a setvar variable for a class variable. See also cab 2012 and df-clab 2013. Note that this is not a rigorous justification, because cv 1232 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 ∈ x} | ||
Definition | df-clel 2022* |
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 2019 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2019 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with setvar variables (see cleljust 1799), so we don't include
any set theory axiom as a hypothesis. See also comments about the
syntax under df-clab 2013.
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 ∈ B ↔ ∃x(x = A ∧ x ∈ B)) | ||
Theorem | eqriv 2023* | Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (x ∈ A ↔ x ∈ B) ⇒ ⊢ A = B | ||
Theorem | eqrdv 2024* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |
⊢ (φ → (x ∈ A ↔ x ∈ B)) ⇒ ⊢ (φ → A = B) | ||
Theorem | eqrdav 2025* | 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 ∈ A ↔ x ∈ B)) ⇒ ⊢ (φ → A = B) | ||
Theorem | eqid 2026 |
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: "Therefore, inquiring why a thing is itself, it's inquiring nothing; ... saying that the thing is itself constitutes the sole reasoning and the sole cause, in every case, to the question of why the man is man or the musician musician."). (Thanks to Stefan Allan and Benoît Jubin for this information.) (Contributed by NM, 5-Aug-1993.) (Revised by Benoît Jubin, 14-Oct-2017.) |
⊢ A = A | ||
Theorem | eqidd 2027 | Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) |
⊢ (φ → A = A) | ||
Theorem | eqcom 2028 | Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B ↔ B = A) | ||
Theorem | eqcoms 2029 | Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B → φ) ⇒ ⊢ (B = A → φ) | ||
Theorem | eqcomi 2030 | Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B ⇒ ⊢ B = A | ||
Theorem | eqcomd 2031 | Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → B = A) | ||
Theorem | eqeq1 2032 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B → (A = 𝐶 ↔ B = 𝐶)) | ||
Theorem | eqeq1i 2033 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B ⇒ ⊢ (A = 𝐶 ↔ B = 𝐶) | ||
Theorem | eqeq1d 2034 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → (A = 𝐶 ↔ B = 𝐶)) | ||
Theorem | eqeq2 2035 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B → (𝐶 = A ↔ 𝐶 = B)) | ||
Theorem | eqeq2i 2036 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B ⇒ ⊢ (𝐶 = A ↔ 𝐶 = B) | ||
Theorem | eqeq2d 2037 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → (𝐶 = A ↔ 𝐶 = B)) | ||
Theorem | eqeq12 2038 | Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.) |
⊢ ((A = B ∧ 𝐶 = 𝐷) → (A = 𝐶 ↔ B = 𝐷)) | ||
Theorem | eqeq12i 2039 | 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) | ||
Theorem | eqeq12d 2040 | A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → A = B) & ⊢ (φ → 𝐶 = 𝐷) ⇒ ⊢ (φ → (A = 𝐶 ↔ B = 𝐷)) | ||
Theorem | eqeqan12d 2041 | A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → A = B) & ⊢ (ψ → 𝐶 = 𝐷) ⇒ ⊢ ((φ ∧ ψ) → (A = 𝐶 ↔ B = 𝐷)) | ||
Theorem | eqeqan12rd 2042 | 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 | eqtr 2043 | 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 2044 | 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) | ||
Theorem | eqtr3 2045 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) |
⊢ ((A = 𝐶 ∧ B = 𝐶) → A = B) | ||
Theorem | eqtri 2046 | An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B & ⊢ B = 𝐶 ⇒ ⊢ A = 𝐶 | ||
Theorem | eqtr2i 2047 | An equality transitivity inference. (Contributed by NM, 21-Feb-1995.) |
⊢ A = B & ⊢ B = 𝐶 ⇒ ⊢ 𝐶 = A | ||
Theorem | eqtr3i 2048 | An equality transitivity inference. (Contributed by NM, 6-May-1994.) |
⊢ A = B & ⊢ A = 𝐶 ⇒ ⊢ B = 𝐶 | ||
Theorem | eqtr4i 2049 | An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B & ⊢ 𝐶 = B ⇒ ⊢ A = 𝐶 | ||
Theorem | 3eqtri 2050 | An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.) |
⊢ A = B & ⊢ B = 𝐶 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ A = 𝐷 | ||
Theorem | 3eqtrri 2051 | 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 2052 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) |
⊢ A = B & ⊢ 𝐶 = B & ⊢ 𝐶 = 𝐷 ⇒ ⊢ A = 𝐷 | ||
Theorem | 3eqtr2ri 2053 | 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 2054 | 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 2055 | An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.) |
⊢ A = B & ⊢ A = 𝐶 & ⊢ B = 𝐷 ⇒ ⊢ 𝐷 = 𝐶 | ||
Theorem | 3eqtr4i 2056 | 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 | ||
Theorem | 3eqtr4ri 2057 | 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 2058 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → A = B) & ⊢ (φ → B = 𝐶) ⇒ ⊢ (φ → A = 𝐶) | ||
Theorem | eqtr2d 2059 | An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.) |
⊢ (φ → A = B) & ⊢ (φ → B = 𝐶) ⇒ ⊢ (φ → 𝐶 = A) | ||
Theorem | eqtr3d 2060 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
⊢ (φ → A = B) & ⊢ (φ → A = 𝐶) ⇒ ⊢ (φ → B = 𝐶) | ||
Theorem | eqtr4d 2061 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
⊢ (φ → A = B) & ⊢ (φ → 𝐶 = B) ⇒ ⊢ (φ → A = 𝐶) | ||
Theorem | 3eqtrd 2062 | 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 2063 | 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 2064 | 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 2065 | 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 2066 | 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 2067 | A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.) |
⊢ (φ → A = B) & ⊢ (φ → A = 𝐶) & ⊢ (φ → B = 𝐷) ⇒ ⊢ (φ → 𝐷 = 𝐶) | ||
Theorem | 3eqtr4d 2068 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → A = B) & ⊢ (φ → 𝐶 = A) & ⊢ (φ → 𝐷 = B) ⇒ ⊢ (φ → 𝐶 = 𝐷) | ||
Theorem | 3eqtr4rd 2069 | A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.) |
⊢ (φ → A = B) & ⊢ (φ → 𝐶 = A) & ⊢ (φ → 𝐷 = B) ⇒ ⊢ (φ → 𝐷 = 𝐶) | ||
Theorem | syl5eq 2070 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B & ⊢ (φ → B = 𝐶) ⇒ ⊢ (φ → A = 𝐶) | ||
Theorem | syl5req 2071 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ A = B & ⊢ (φ → B = 𝐶) ⇒ ⊢ (φ → 𝐶 = A) | ||
Theorem | syl5eqr 2072 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ B = A & ⊢ (φ → B = 𝐶) ⇒ ⊢ (φ → A = 𝐶) | ||
Theorem | syl5reqr 2073 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ B = A & ⊢ (φ → B = 𝐶) ⇒ ⊢ (φ → 𝐶 = A) | ||
Theorem | syl6eq 2074 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → A = B) & ⊢ B = 𝐶 ⇒ ⊢ (φ → A = 𝐶) | ||
Theorem | syl6req 2075 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ (φ → A = B) & ⊢ B = 𝐶 ⇒ ⊢ (φ → 𝐶 = A) | ||
Theorem | syl6eqr 2076 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → A = B) & ⊢ 𝐶 = B ⇒ ⊢ (φ → A = 𝐶) | ||
Theorem | syl6reqr 2077 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ (φ → A = B) & ⊢ 𝐶 = B ⇒ ⊢ (φ → 𝐶 = A) | ||
Theorem | sylan9eq 2078 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → A = B) & ⊢ (ψ → B = 𝐶) ⇒ ⊢ ((φ ∧ ψ) → A = 𝐶) | ||
Theorem | sylan9req 2079 | An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.) |
⊢ (φ → B = A) & ⊢ (ψ → B = 𝐶) ⇒ ⊢ ((φ ∧ ψ) → A = 𝐶) | ||
Theorem | sylan9eqr 2080 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) |
⊢ (φ → A = B) & ⊢ (ψ → B = 𝐶) ⇒ ⊢ ((ψ ∧ φ) → A = 𝐶) | ||
Theorem | 3eqtr3g 2081 | A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.) |
⊢ (φ → A = B) & ⊢ A = 𝐶 & ⊢ B = 𝐷 ⇒ ⊢ (φ → 𝐶 = 𝐷) | ||
Theorem | 3eqtr3a 2082 | A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.) |
⊢ A = B & ⊢ (φ → A = 𝐶) & ⊢ (φ → B = 𝐷) ⇒ ⊢ (φ → 𝐶 = 𝐷) | ||
Theorem | 3eqtr4g 2083 | A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → A = B) & ⊢ 𝐶 = A & ⊢ 𝐷 = B ⇒ ⊢ (φ → 𝐶 = 𝐷) | ||
Theorem | 3eqtr4a 2084 | 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 & ⊢ (φ → 𝐶 = A) & ⊢ (φ → 𝐷 = B) ⇒ ⊢ (φ → 𝐶 = 𝐷) | ||
Theorem | eq2tri 2085 | 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)) | ||
Theorem | eleq1 2086 | Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B → (A ∈ 𝐶 ↔ B ∈ 𝐶)) | ||
Theorem | eleq2 2087 | Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B → (𝐶 ∈ A ↔ 𝐶 ∈ B)) | ||
Theorem | eleq12 2088 | Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.) |
⊢ ((A = B ∧ 𝐶 = 𝐷) → (A ∈ 𝐶 ↔ B ∈ 𝐷)) | ||
Theorem | eleq1i 2089 | Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B ⇒ ⊢ (A ∈ 𝐶 ↔ B ∈ 𝐶) | ||
Theorem | eleq2i 2090 | Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B ⇒ ⊢ (𝐶 ∈ A ↔ 𝐶 ∈ B) | ||
Theorem | eleq12i 2091 | Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
⊢ A = B & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (A ∈ 𝐶 ↔ B ∈ 𝐷) | ||
Theorem | eleq1d 2092 | Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → (A ∈ 𝐶 ↔ B ∈ 𝐶)) | ||
Theorem | eleq2d 2093 | Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → (𝐶 ∈ A ↔ 𝐶 ∈ B)) | ||
Theorem | eleq12d 2094 | Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
⊢ (φ → A = B) & ⊢ (φ → 𝐶 = 𝐷) ⇒ ⊢ (φ → (A ∈ 𝐶 ↔ B ∈ 𝐷)) | ||
Theorem | eleq1a 2095 | A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.) |
⊢ (A ∈ B → (𝐶 = A → 𝐶 ∈ B)) | ||
Theorem | eqeltri 2096 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B & ⊢ B ∈ 𝐶 ⇒ ⊢ A ∈ 𝐶 | ||
Theorem | eqeltrri 2097 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B & ⊢ A ∈ 𝐶 ⇒ ⊢ B ∈ 𝐶 | ||
Theorem | eleqtri 2098 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ A ∈ B & ⊢ B = 𝐶 ⇒ ⊢ A ∈ 𝐶 | ||
Theorem | eleqtrri 2099 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ A ∈ B & ⊢ 𝐶 = B ⇒ ⊢ A ∈ 𝐶 | ||
Theorem | eqeltrd 2100 | Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ (φ → A = B) & ⊢ (φ → B ∈ 𝐶) ⇒ ⊢ (φ → A ∈ 𝐶) |
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