P. 23 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bamalip 2201	"Bamalip", one of the syllogisms of Aristotelian logic. All ph is ps, all ps is ch, and ph exist, therefore some ch is ph. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2182. (Contributed by David A. Wheeler, 28-Aug-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ps -> ch )   &    |- E. x ph   =>    |- E. x ( ch /\ ph )
1.5  First-order logic with one non-logical binary predicate This section adds one non-logical binary predicate to the first-order logic developed until here. We call it "the membership predicate" since it will be used in the next part as the membership predicate of set theory, but in this section, it has no other property than being "a binary predicate". "Non-logical" means that it does not belong to the logic. In our logic (and in most treatments), the only logical predicate is the equality predicate (see weq 1551).
Syntax	wcel 2202	Extend wff definition to include the membership connective between classes. For a general discussion of the theory of classes, see mmset.html#class. The purpose of introducing wff A e. B here is to allow us to prove the wel 2203 of predicate calculus in terms of the wceq 1397 of set theory, so that we do not overload the e. connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2218 for more information on the set theory usage of wcel 2202.
wff A e. B
Theorem	wel 2203	Extend wff definition to include atomic formulas with the membership predicate. This is read either " x is an element of y", or " x is a member of y", or " x belongs to y", or " y contains x". Note: The phrase " y includes x " means " x is a subset of y"; to use it also for x e. y, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT). This syntactical construction introduces a binary non-logical predicate symbol e. into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for e. apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. Instead of introducing wel 2203 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 2202. This lets us avoid overloading the e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 2203 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 2202. Note: To see the proof steps of this syntax proof, type "MM> SHOW PROOF wel / ALL" in the Metamath program. (Contributed by NM, 24-Jan-2006.)
wff x e. y
Axiom	ax-13 2204	Axiom of left equality for the binary predicate e.. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the e. binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x e. z -> y e. z ) )
Axiom	ax-14 2205	Axiom of right equality for the binary predicate e.. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the e. binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z e. x -> z e. y ) )
Theorem	elequ1 2206	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x e. z <-> y e. z ) )
Theorem	elequ2 2207	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z e. x <-> z e. y ) )
Theorem	cleljust 2208*	When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2203 with the class variables in wcel 2202. (Contributed by NM, 28-Jan-2004.)
|- ( x e. y <-> E. z ( z = x /\ z e. y ) )
Theorem	elsb1 2209*	Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2210 for substitution for the second argument. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] x e. z <-> y e. z )
Theorem	elsb2 2210*	Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2209 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] z e. x <-> z e. y )
Theorem	dveel1 2211*	Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. x x = y -> ( y e. z -> A. x y e. z ) )
Theorem	dveel2 2212*	Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) )
PART 2  SET THEORY Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets". A set can be an element of another set, and this relationship is indicated by the e. symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. Here we develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in Metamath because the statement of some of its axioms uses the notion of "bounded formula". Since Metamath has, purposefully, a very weak metalogic, that notion must be developed in the logic itself. This is similar to our treatment of substitution (df-sb 1811) and our definition of the nonfreeness predicate (df-nf 1509), whereas substitution and bound and free variables are ordinarily defined in the metalogic. The development of CZF has begun in BJ's mathbox, see wbd 16407.
2.1  IZF Set Theory - start with the Axiom of Extensionality
2.1.1  Introduce the Axiom of Extensionality
Axiom	ax-ext 2213*	Axiom of Extensionality. It states that two sets are identical if they contain the same elements. Axiom 1 of [Crosilla] p. "Axioms of CZF and IZF" (with unnecessary quantifiers removed). Set theory can also be formulated with a single primitive predicate e. on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes ( A. w ( w e. x <-> w e. y ) -> ( x e. z -> y e. z ) ), and equality x = y is defined as A. w ( w e. x <-> w e. y ). All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8. To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-8 1552 through ax-16 1862 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 2213 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 2213 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.)
|- ( A. z ( z e. x <-> z e. y ) -> x = y )
Theorem	axext3 2214*	A generalization of the Axiom of Extensionality in which x and y need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- ( A. z ( z e. x <-> z e. y ) -> x = y )
Theorem	axext4 2215*	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 2213. (Contributed by NM, 14-Nov-2008.)
|- ( x = y <-> A. z ( z e. x <-> z e. y ) )
Theorem	bm1.1 2216*	Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.)
|- F/ x ph   =>    |- ( E. x A. y ( y e. x <-> ph ) -> E! x A. y ( y e. x <-> ph ) )
2.1.2  Class abstractions (a.k.a. class builders)
Syntax	cab 2217	Introduce the class builder or class abstraction notation ("the class of sets x such that ph is true"). Our class variables A, B, etc. range over class builders (sometimes implicitly). Note that a setvar variable can be expressed as a class builder per Theorem cvjust 2226, justifying the assignment of setvar variables to class variables via the use of cv 1396.
class { x | ph }
Definition	df-clab 2218	Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature. x and y need not be distinct. Definition 2.1 of [Quine] p. 16. Typically, ph will have y as a free variable, and " { y | ph } " is read "the class of all sets y such that ph ( y ) is true". We do not define { y | ph } in isolation but only as part of an expression that extends or "overloads" the e. relationship. This is our first use of the e. symbol to connect classes instead of sets. The syntax definition wcel 2202, which extends or "overloads" the wel 2203 definition connecting setvar variables, requires that both sides of e. be a class. In df-cleq 2224 and df-clel 2227, we introduce a new kind of variable (class variable) that can substituted with expressions such as { y | ph }. In the present definition, the x on the left-hand side is a setvar variable. Syntax definition cv 1396 allows us to substitute a setvar variable x for a class variable: all sets are classes by cvjust 2226 (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 2340 for a quick overview). Because class variables can be substituted with compound expressions and setvar variables cannot, it is often useful to convert a theorem containing a free setvar variable to a more general version with a class variable. This is called the "axiom of class comprehension" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. He calls the construction { y | ph } a "class term". For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class 2340. (Contributed by NM, 5-Aug-1993.)
|- ( x e. { y | ph } <-> [ x / y ] ph )
Theorem	abid 2219	Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.)
|- ( x e. { x | ph } <-> ph )
Theorem	hbab1 2220*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)
|- ( y e. { x | ph } -> A. x y e. { x | ph } )
Theorem	nfsab1 2221*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x y e. { x | ph }
Theorem	hbab 2222*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
|- ( ph -> A. x ph )   =>    |- ( z e. { y | ph } -> A. x z e. { y | ph } )
Theorem	nfsab 2223*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- F/ x z e. { y | ph }
Definition	df-cleq 2224*	Define the equality connective between classes. Definition 2.7 of [Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4 provides its justification and methods for eliminating it. Note that its elimination will not necessarily result in a single wff in the original language but possibly a "scheme" of wffs. This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce y = z <-> A. x ( x e. y <-> x e. z ), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see Theorem axext4 2215). 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 2218, df-clel 2227, and abeq2 2340. In the form of dfcleq 2225, 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 2225. (Contributed by NM, 15-Sep-1993.)
|- ( A. x ( x e. y <-> x e. z ) -> y = z )   =>    |- ( A = B <-> A. x ( x e. A <-> x e. B ) )
Theorem	dfcleq 2225*	The same as df-cleq 2224 with the hypothesis removed using the Axiom of Extensionality ax-ext 2213. (Contributed by NM, 15-Sep-1993.)
|- ( A = B <-> A. x ( x e. A <-> x e. B ) )
Theorem	cvjust 2226*	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 1396, which allows us to substitute a setvar variable for a class variable. See also cab 2217 and df-clab 2218. Note that this is not a rigorous justification, because cv 1396 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." (Contributed by NM, 7-Nov-2006.)
|- x = { y | y e. x }
Definition	df-clel 2227*	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 2224 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2224 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2208), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2218. 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 2218. (Contributed by NM, 5-Aug-1993.)
|- ( A e. B <-> E. x ( x = A /\ x e. B ) )
Theorem	eqriv 2228*	Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- ( x e. A <-> x e. B )   =>    |- A = B
Theorem	eqrdv 2229*	Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.)
|- ( ph -> ( x e. A <-> x e. B ) )   =>    |- ( ph -> A = B )
Theorem	eqrdav 2230*	Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.)
|- ( ( ph /\ x e. A ) -> x e. C )   &    |- ( ( ph /\ x e. B ) -> x e. C )   &    |- ( ( ph /\ x e. C ) -> ( x e. A <-> x e. B ) )   =>    |- ( ph -> A = B )
Theorem	eqid 2231	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 2232	Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
|- ( ph -> A = A )
Theorem	eqcom 2233	Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.)
|- ( A = B <-> B = A )
Theorem	eqcoms 2234	Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ph )   =>    |- ( B = A -> ph )
Theorem	eqcomi 2235	Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- B = A
Theorem	neqcomd 2236	Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> -. A = B )   =>    |- ( ph -> -. B = A )
Theorem	eqcomd 2237	Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.)
|- ( ph -> A = B )   =>    |- ( ph -> B = A )
Theorem	eqeq1 2238	Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( A = C <-> B = C ) )
Theorem	eqeq1i 2239	Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- ( A = C <-> B = C )
Theorem	eqeq1d 2240	Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A = C <-> B = C ) )
Theorem	eqeq2 2241	Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( C = A <-> C = B ) )
Theorem	eqeq2i 2242	Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- ( C = A <-> C = B )
Theorem	eqeq2d 2243	Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C = A <-> C = B ) )
Theorem	eqeq12 2244	Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
|- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )
Theorem	eqeq12i 2245	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 2246	A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A = C <-> B = D ) )
Theorem	eqeqan12d 2247	A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )
Theorem	eqeqan12rd 2248	A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ps /\ ph ) -> ( A = C <-> B = D ) )
Theorem	eqtr 2249	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 2250	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 2251	A transitive law for class equality. (Contributed by NM, 20-May-2005.)
|- ( ( A = C /\ B = C ) -> A = B )
Theorem	eqtri 2252	An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- B = C   =>    |- A = C
Theorem	eqtr2i 2253	An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
|- A = B   &    |- B = C   =>    |- C = A
Theorem	eqtr3i 2254	An equality transitivity inference. (Contributed by NM, 6-May-1994.)
|- A = B   &    |- A = C   =>    |- B = C
Theorem	eqtr4i 2255	An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- C = B   =>    |- A = C
Theorem	3eqtri 2256	An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
|- A = B   &    |- B = C   &    |- C = D   =>    |- A = D
Theorem	3eqtrri 2257	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 2258	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
|- A = B   &    |- C = B   &    |- C = D   =>    |- A = D
Theorem	3eqtr2ri 2259	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 2260	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 2261	An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
|- A = B   &    |- A = C   &    |- B = D   =>    |- D = C
Theorem	3eqtr4i 2262	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 2263	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 2264	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A = B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A = C )
Theorem	eqtr2d 2265	An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
|- ( ph -> A = B )   &    |- ( ph -> B = C )   =>    |- ( ph -> C = A )
Theorem	eqtr3d 2266	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
|- ( ph -> A = B )   &    |- ( ph -> A = C )   =>    |- ( ph -> B = C )
Theorem	eqtr4d 2267	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
|- ( ph -> A = B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A = C )
Theorem	3eqtrd 2268	A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
|- ( ph -> A = B )   &    |- ( ph -> B = C )   &    |- ( ph -> C = D )   =>    |- ( ph -> A = D )
Theorem	3eqtrrd 2269	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ph -> B = C )   &    |- ( ph -> C = D )   =>    |- ( ph -> D = A )
Theorem	3eqtr2d 2270	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
|- ( ph -> A = B )   &    |- ( ph -> C = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> A = D )
Theorem	3eqtr2rd 2271	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
|- ( ph -> A = B )   &    |- ( ph -> C = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> D = A )
Theorem	3eqtr3d 2272	A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C = D )
Theorem	3eqtr3rd 2273	A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
|- ( ph -> A = B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> D = C )
Theorem	3eqtr4d 2274	A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C = D )
Theorem	3eqtr4rd 2275	A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
|- ( ph -> A = B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> D = C )
Theorem	eqtrid 2276	An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
|- A = B   &    |- ( ph -> B = C )   =>    |- ( ph -> A = C )
Theorem	eqtr2id 2277	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- A = B   &    |- ( ph -> B = C )   =>    |- ( ph -> C = A )
Theorem	eqtr3id 2278	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
|- B = A   &    |- ( ph -> B = C )   =>    |- ( ph -> A = C )
Theorem	eqtr3di 2279	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- ( ph -> A = B )   &    |- A = C   =>    |- ( ph -> B = C )
Theorem	eqtrdi 2280	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A = B )   &    |- B = C   =>    |- ( ph -> A = C )
Theorem	eqtr2di 2281	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- ( ph -> A = B )   &    |- B = C   =>    |- ( ph -> C = A )
Theorem	eqtr4di 2282	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A = B )   &    |- C = B   =>    |- ( ph -> A = C )
Theorem	eqtr4id 2283	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- A = B   &    |- ( ph -> C = B )   =>    |- ( ph -> A = C )
Theorem	sylan9eq 2284	An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ps -> B = C )   =>    |- ( ( ph /\ ps ) -> A = C )
Theorem	sylan9req 2285	An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
|- ( ph -> B = A )   &    |- ( ps -> B = C )   =>    |- ( ( ph /\ ps ) -> A = C )
Theorem	sylan9eqr 2286	An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
|- ( ph -> A = B )   &    |- ( ps -> B = C )   =>    |- ( ( ps /\ ph ) -> A = C )
Theorem	3eqtr3g 2287	A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
|- ( ph -> A = B )   &    |- A = C   &    |- B = D   =>    |- ( ph -> C = D )
Theorem	3eqtr3a 2288	A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
|- A = B   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C = D )
Theorem	3eqtr4g 2289	A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A = B )   &    |- C = A   &    |- D = B   =>    |- ( ph -> C = D )
Theorem	3eqtr4a 2290	A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C = D )
Theorem	eq2tri 2291	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 2292	Weaker version of eleq1 2294 (but more general than elequ1 2206) not depending on ax-ext 2213 nor df-cleq 2224. (Contributed by BJ, 24-Jun-2019.)
|- ( x = y -> ( x e. A <-> y e. A ) )
Theorem	eleq2w 2293	Weaker version of eleq2 2295 (but more general than elequ2 2207) not depending on ax-ext 2213 nor df-cleq 2224. (Contributed by BJ, 29-Sep-2019.)
|- ( x = y -> ( A e. x <-> A e. y ) )
Theorem	eleq1 2294	Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( A e. C <-> B e. C ) )
Theorem	eleq2 2295	Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( C e. A <-> C e. B ) )
Theorem	eleq12 2296	Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
|- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) )
Theorem	eleq1i 2297	Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- ( A e. C <-> B e. C )
Theorem	eleq2i 2298	Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- ( C e. A <-> C e. B )
Theorem	eleq12i 2299	Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
|- A = B   &    |- C = D   =>    |- ( A e. C <-> B e. D )
Theorem	eleq1d 2300	Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A e. C <-> B e. C ) )