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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cab 2101 | 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 2110, justifying the assignment of setvar variables to class variables via the use of cv 1313. |
class {𝑥 ∣ 𝜑} | ||
Definition | df-clab 2102 |
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 1463, which extends or "overloads" the wel 1464 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2108 and df-clel 2111, 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 1313 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2110 (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 2224 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 https://us.metamath.org/mpeuni/mmset.html#class 2224. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) | ||
Theorem | abid 2103 | 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 2104* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | nfsab1 2105* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | ||
Theorem | hbab 2106* | 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 2107* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑} | ||
Definition | df-cleq 2108* |
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 2099). 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 2102, df-clel 2111, and abeq2 2224. In the form of dfcleq 2109, 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 2109. (Contributed by NM, 15-Sep-1993.) |
⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | dfcleq 2109* | The same as df-cleq 2108 with the hypothesis removed using the Axiom of Extensionality ax-ext 2097. (Contributed by NM, 15-Sep-1993.) |
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | cvjust 2110* | 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 1313, which allows us to substitute a setvar variable for a class variable. See also cab 2101 and df-clab 2102. Note that this is not a rigorous justification, because cv 1313 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 2111* |
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 2108 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2108 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with setvar variables (see cleljust 1888), so we don't include
any set theory axiom as a hypothesis. See also comments about the
syntax under df-clab 2102.
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 2102. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) | ||
Theorem | eqriv 2112* | Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ⇒ ⊢ 𝐴 = 𝐵 | ||
Theorem | eqrdv 2113* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | eqrdav 2114* | 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 2115 |
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 | ||
Theorem | eqidd 2116 | Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) |
⊢ (𝜑 → 𝐴 = 𝐴) | ||
Theorem | eqcom 2117 | Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | ||
Theorem | eqcoms 2118 | Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → 𝜑) ⇒ ⊢ (𝐵 = 𝐴 → 𝜑) | ||
Theorem | eqcomi 2119 | Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝐵 = 𝐴 | ||
Theorem | neqcomd 2120 | Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 = 𝐴) | ||
Theorem | eqcomd 2121 | Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐵 = 𝐴) | ||
Theorem | eqeq1 2122 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | ||
Theorem | eqeq1i 2123 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐶) | ||
Theorem | eqeq1d 2124 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | ||
Theorem | eqeq2 2125 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)) | ||
Theorem | eqeq2i 2126 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵) | ||
Theorem | eqeq2d 2127 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)) | ||
Theorem | eqeq12 2128 | Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqeq12i 2129 | 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 2130 | 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 2131 | 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 2132 | 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 2133 | 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 2134 | 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 2135 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) |
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵) | ||
Theorem | eqtri 2136 | An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴 = 𝐶 | ||
Theorem | eqtr2i 2137 | An equality transitivity inference. (Contributed by NM, 21-Feb-1995.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐶 = 𝐴 | ||
Theorem | eqtr3i 2138 | An equality transitivity inference. (Contributed by NM, 6-May-1994.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 = 𝐶 ⇒ ⊢ 𝐵 = 𝐶 | ||
Theorem | eqtr4i 2139 | An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴 = 𝐶 | ||
Theorem | 3eqtri 2140 | An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ 𝐴 = 𝐷 | ||
Theorem | 3eqtrri 2141 | 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 2142 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ 𝐴 = 𝐷 | ||
Theorem | 3eqtr2ri 2143 | 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 2144 | 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 2145 | An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐷 = 𝐶 | ||
Theorem | 3eqtr4i 2146 | 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 2147 | 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 2148 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | eqtr2d 2149 | An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | eqtr3d 2150 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
Theorem | eqtr4d 2151 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | 3eqtrd 2152 | 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 2153 | 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 2154 | 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 2155 | 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 2156 | 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 2157 | 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 2158 | 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 2159 | 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 2160 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl5req 2161 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | syl5eqr 2162 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl5reqr 2163 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | syl6eq 2164 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl6req 2165 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | syl6eqr 2166 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl6reqr 2167 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | sylan9eq 2168 | 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 2169 | An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ (𝜓 → 𝐵 = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) | ||
Theorem | sylan9eqr 2170 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐵 = 𝐶) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝐴 = 𝐶) | ||
Theorem | 3eqtr3g 2171 | 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 2172 | 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 2173 | 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 2174 | 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 2175 | 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 | eleq1w 2176 | Weaker version of eleq1 2178 (but more general than elequ1 1673) not depending on ax-ext 2097 nor df-cleq 2108. (Contributed by BJ, 24-Jun-2019.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | ||
Theorem | eleq2w 2177 | Weaker version of eleq2 2179 (but more general than elequ2 1674) not depending on ax-ext 2097 nor df-cleq 2108. (Contributed by BJ, 29-Sep-2019.) |
⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) | ||
Theorem | eleq1 2178 | Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | ||
Theorem | eleq2 2179 | Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | ||
Theorem | eleq12 2180 | Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | ||
Theorem | eleq1i 2181 | Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶) | ||
Theorem | eleq2i 2182 | Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵) | ||
Theorem | eleq12i 2183 | Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷) | ||
Theorem | eleq1d 2184 | Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | ||
Theorem | eleq2d 2185 | Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | ||
Theorem | eleq12d 2186 | 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 2187 | A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.) |
⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵)) | ||
Theorem | eqeltri 2188 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ 𝐴 ∈ 𝐶 | ||
Theorem | eqeltrri 2189 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ 𝐵 ∈ 𝐶 | ||
Theorem | eleqtri 2190 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴 ∈ 𝐶 | ||
Theorem | eleqtrri 2191 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴 ∈ 𝐶 | ||
Theorem | eqeltrd 2192 | 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) | ||
Theorem | eqeltrrd 2193 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝐶) | ||
Theorem | eleqtrd 2194 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eleqtrrd 2195 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | 3eltr3i 2196 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐶 ∈ 𝐷 | ||
Theorem | 3eltr4i 2197 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ 𝐶 ∈ 𝐷 | ||
Theorem | 3eltr3d 2198 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) | ||
Theorem | 3eltr4d 2199 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐴) & ⊢ (𝜑 → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) | ||
Theorem | 3eltr3g 2200 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
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