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Theorem List for Intuitionistic Logic Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxext4 2101* 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 2099. (Contributed by NM, 14-Nov-2008.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
 
Theorembm1.1 2102* 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(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑))
 
2.1.2  Class abstractions (a.k.a. class builders)
 
Syntaxcab 2103 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 2112, justifying the assignment of setvar variables to class variables via the use of cv 1315.
class {𝑥𝜑}
 
Definitiondf-clab 2104 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 1465, which extends or "overloads" the wel 1466 definition connecting setvar variables, requires that both sides of be a class. In df-cleq 2110 and df-clel 2113, 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 1315 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2112 (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 2226 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 https://us.metamath.org/mpeuni/mmset.html#class 2226. (Contributed by NM, 5-Aug-1993.)

(𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
 
Theoremabid 2105 Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.)
(𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
 
Theoremhbab1 2106* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)
(𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
 
Theoremnfsab1 2107* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥 𝑦 ∈ {𝑥𝜑}
 
Theoremhbab 2108* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
(𝜑 → ∀𝑥𝜑)       (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
 
Theoremnfsab 2109* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥 𝑧 ∈ {𝑦𝜑}
 
Definitiondf-cleq 2110* 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(𝑥𝑦𝑥𝑧), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2101). 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.

See also comments under df-clab 2104, df-clel 2113, and abeq2 2226.

In the form of dfcleq 2111, 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 2111. (Contributed by NM, 15-Sep-1993.)

(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremdfcleq 2111* The same as df-cleq 2110 with the hypothesis removed using the Axiom of Extensionality ax-ext 2099. (Contributed by NM, 15-Sep-1993.)
(𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremcvjust 2112* 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 1315, which allows us to substitute a setvar variable for a class variable. See also cab 2103 and df-clab 2104. Note that this is not a rigorous justification, because cv 1315 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 = {𝑦𝑦𝑥}
 
Definitiondf-clel 2113* 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 2110 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2110 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1890), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2104.

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 2104. (Contributed by NM, 5-Aug-1993.)

(𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
 
Theoremeqriv 2114* Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝑥𝐴𝑥𝐵)       𝐴 = 𝐵
 
Theoremeqrdv 2115* Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.)
(𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremeqrdav 2116* Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.)
((𝜑𝑥𝐴) → 𝑥𝐶)    &   ((𝜑𝑥𝐵) → 𝑥𝐶)    &   ((𝜑𝑥𝐶) → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremeqid 2117 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 2118 Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
(𝜑𝐴 = 𝐴)
 
Theoremeqcom 2119 Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵𝐵 = 𝐴)
 
Theoremeqcoms 2120 Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵𝜑)       (𝐵 = 𝐴𝜑)
 
Theoremeqcomi 2121 Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       𝐵 = 𝐴
 
Theoremneqcomd 2122 Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑 → ¬ 𝐵 = 𝐴)
 
Theoremeqcomd 2123 Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑𝐵 = 𝐴)
 
Theoremeqeq1 2124 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
 
Theoremeqeq1i 2125 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       (𝐴 = 𝐶𝐵 = 𝐶)
 
Theoremeqeq1d 2126 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐶))
 
Theoremeqeq2 2127 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐶 = 𝐴𝐶 = 𝐵))
 
Theoremeqeq2i 2128 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       (𝐶 = 𝐴𝐶 = 𝐵)
 
Theoremeqeq2d 2129 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶 = 𝐴𝐶 = 𝐵))
 
Theoremeqeq12 2130 Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremeqeq12i 2131 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 2132 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 2133 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)       ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremeqeqan12rd 2134 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremeqtr 2135 Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)
((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)
 
Theoremeqtr2 2136 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
((𝐴 = 𝐵𝐴 = 𝐶) → 𝐵 = 𝐶)
 
Theoremeqtr3 2137 A transitive law for class equality. (Contributed by NM, 20-May-2005.)
((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
 
Theoremeqtri 2138 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐵 = 𝐶       𝐴 = 𝐶
 
Theoremeqtr2i 2139 An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
𝐴 = 𝐵    &   𝐵 = 𝐶       𝐶 = 𝐴
 
Theoremeqtr3i 2140 An equality transitivity inference. (Contributed by NM, 6-May-1994.)
𝐴 = 𝐵    &   𝐴 = 𝐶       𝐵 = 𝐶
 
Theoremeqtr4i 2141 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐶 = 𝐵       𝐴 = 𝐶
 
Theorem3eqtri 2142 An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
𝐴 = 𝐵    &   𝐵 = 𝐶    &   𝐶 = 𝐷       𝐴 = 𝐷
 
Theorem3eqtrri 2143 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 2144 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
𝐴 = 𝐵    &   𝐶 = 𝐵    &   𝐶 = 𝐷       𝐴 = 𝐷
 
Theorem3eqtr2ri 2145 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 2146 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 2147 An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
𝐴 = 𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐷 = 𝐶
 
Theorem3eqtr4i 2148 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 2149 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 2150 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)
 
Theoremeqtr2d 2151 An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐶 = 𝐴)
 
Theoremeqtr3d 2152 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴 = 𝐶)       (𝜑𝐵 = 𝐶)
 
Theoremeqtr4d 2153 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴 = 𝐶)
 
Theorem3eqtrd 2154 A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐴 = 𝐷)
 
Theorem3eqtrrd 2155 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 2156 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐴 = 𝐷)
 
Theorem3eqtr2rd 2157 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐷 = 𝐴)
 
Theorem3eqtr3d 2158 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 2159 A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐷 = 𝐶)
 
Theorem3eqtr4d 2160 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 2161 A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐷 = 𝐶)
 
Theoremsyl5eq 2162 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)
 
Theoremsyl5req 2163 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
𝐴 = 𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐶 = 𝐴)
 
Theoremsyl5eqr 2164 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
𝐵 = 𝐴    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)
 
Theoremsyl5reqr 2165 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
𝐵 = 𝐴    &   (𝜑𝐵 = 𝐶)       (𝜑𝐶 = 𝐴)
 
Theoremsyl6eq 2166 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)    &   𝐵 = 𝐶       (𝜑𝐴 = 𝐶)
 
Theoremsyl6req 2167 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)    &   𝐵 = 𝐶       (𝜑𝐶 = 𝐴)
 
Theoremsyl6eqr 2168 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)    &   𝐶 = 𝐵       (𝜑𝐴 = 𝐶)
 
Theoremsyl6reqr 2169 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)    &   𝐶 = 𝐵       (𝜑𝐶 = 𝐴)
 
Theoremsylan9eq 2170 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐵 = 𝐶)       ((𝜑𝜓) → 𝐴 = 𝐶)
 
Theoremsylan9req 2171 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
(𝜑𝐵 = 𝐴)    &   (𝜓𝐵 = 𝐶)       ((𝜑𝜓) → 𝐴 = 𝐶)
 
Theoremsylan9eqr 2172 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐵 = 𝐶)       ((𝜓𝜑) → 𝐴 = 𝐶)
 
Theorem3eqtr3g 2173 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
(𝜑𝐴 = 𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶 = 𝐷)
 
Theorem3eqtr3a 2174 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 2175 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶 = 𝐷)
 
Theorem3eqtr4a 2176 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 2177 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
(𝐴 = 𝐶𝐷 = 𝐹)    &   (𝐵 = 𝐷𝐶 = 𝐺)       ((𝐴 = 𝐶𝐵 = 𝐹) ↔ (𝐵 = 𝐷𝐴 = 𝐺))
 
Theoremeleq1w 2178 Weaker version of eleq1 2180 (but more general than elequ1 1675) not depending on ax-ext 2099 nor df-cleq 2110. (Contributed by BJ, 24-Jun-2019.)
(𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
 
Theoremeleq2w 2179 Weaker version of eleq2 2181 (but more general than elequ2 1676) not depending on ax-ext 2099 nor df-cleq 2110. (Contributed by BJ, 29-Sep-2019.)
(𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
 
Theoremeleq1 2180 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremeleq2 2181 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremeleq12 2182 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
 
Theoremeleq1i 2183 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theoremeleq2i 2184 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremeleq12i 2185 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)
 
Theoremeleq1d 2186 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theoremeleq2d 2187 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremeleq12d 2188 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremeleq1a 2189 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
(𝐴𝐵 → (𝐶 = 𝐴𝐶𝐵))
 
Theoremeqeltri 2190 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremeqeltrri 2191 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶
 
Theoremeleqtri 2192 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶
 
Theoremeleqtrri 2193 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶
 
Theoremeqeltrd 2194 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqeltrrd 2195 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)
 
Theoremeleqtrd 2196 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremeleqtrrd 2197 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theorem3eltr3i 2198 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝐷
 
Theorem3eltr4i 2199 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝐷
 
Theorem3eltr3d 2200 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)
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