P. 27 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 2601-2700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axbnd 2601	Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2600 are fairly straightforward consequences of axc9 2302. But in intuitionistic logic, it is not easy to add the extra A. x to axi12 2600 and so we treat the two as separate axioms. (Contributed by Jim Kingdon, 22-Mar-2018.)
|- ( A. z z = x \/ ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) )
PART 2  ZF (ZERMELO-FRAENKEL) 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. A simplistic concept of sets, sometimes called "naive set theory", is vulnerable to a paradox called "Russell's Paradox" (ru 3434), a discovery that revolutionized the foundations of mathematics and logic. Russell's Paradox spawned the development of set theories that countered the paradox, including the ZF set theory that is most widely used and is defined here. Except for Extensionality, the axioms basically say, "given an arbitrary set x (and, in the cases of Replacement and Regularity, provided that an antecedent is satisfied), there exists another set y based on or constructed from it, with the stated properties." (The axiom of Extensionality can also be restated this way as shown by axext2 2603.) The individual axiom links provide more detailed descriptions. We derive the redundant ZF axioms of Separation, Null Set, and Pairing from the others as theorems.
2.1  ZF Set Theory - start with the Axiom of Extensionality
2.1.1  Introduce the Axiom of Extensionality
Axiom	ax-ext 2602*	Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461. 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 predicate calculus axioms, we would rewrite all axioms involving equality with equality expanded according to the above definition. Some of those axioms may be provable from ax-ext and would become redundant, but this hasn't been studied carefully. General remarks: Our set theory axioms are presented using defined connectives ( <->, E., etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives ->, -., A., =, and e.. It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so. 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 2602 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 Replacement ax-rep 4771, which involves a wff metavariable). 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 2602 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 21-May-1993.)
|- ( A. z ( z e. x <-> z e. y ) -> x = y )
Theorem	axext2 2603*	The Axiom of Extensionality (ax-ext 2602) restated so that it postulates the existence of a set z given two arbitrary sets x and y. This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.)
|- E. z ( ( z e. x <-> z e. y ) -> x = y )
Theorem	axext3 2604*	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.) Remove dependencies on ax-10 2019, ax-12 2047, ax-13 2246. (Revised by Wolf Lammen, 9-Dec-2019.)
|- ( A. z ( z e. x <-> z e. y ) -> x = y )
Theorem	axext3ALT 2605*	Alternate proof of axext3 2604, shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A. z ( z e. x <-> z e. y ) -> x = y )
Theorem	axext4 2606*	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 2602 and df-cleq 2615. (Contributed by NM, 14-Nov-2008.)
|- ( x = y <-> A. z ( z e. x <-> z e. y ) )
Theorem	bm1.1 2607*	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.) (Proof shortened by Wolf Lammen, 13-Nov-2019.)
|- 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 2608	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 (implicitly in the case of defined class terms such as df-nul 3916). Note that a setvar variable can be expressed as a class builder per theorem cvjust 2617, justifying the assignment of setvar variables to class variables via the use of cv 1482.
class { x | ph }
Definition	df-clab 2609	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 1990, which extends or "overloads" the wel 1991 definition connecting setvar variables, requires that both sides of e. be classes. In df-cleq 2615 and df-clel 2618, we introduce a new kind of variable (class variable) that can be 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 1482 allows us to substitute a setvar variable x for a class variable: all sets are classes by cvjust 2617 (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 2732 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 done with theorems such as vtoclg 3266 which is used, for example, to convert elirrv 8504 to elirr 8505. 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". While the three class definitions df-clab 2609, df-cleq 2615, and df-clel 2618 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker. For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-1993.)
|- ( x e. { y | ph } <-> [ x / y ] ph )
Theorem	abid 2610	Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 26-May-1993.)
|- ( x e. { x | ph } <-> ph )
Theorem	hbab1 2611*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.)
|- ( y e. { x | ph } -> A. x y e. { x | ph } )
Theorem	nfsab1 2612*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x y e. { x | ph }
Theorem	hbab 2613*	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 2614*	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 2615*	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 2606). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated. See also comments under df-clab 2609, df-clel 2618, and abeq2 2732. In the form of dfcleq 2616, 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. While the three class definitions df-clab 2609, df-cleq 2615, and df-clel 2618 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker. For a general discussion of the theory of classes, see mmset.html#class. (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 2616*	The same as df-cleq 2615 with the hypothesis removed using the Axiom of Extensionality ax-ext 2602. (Contributed by NM, 15-Sep-1993.) Revised to make use of axext3 2604 instead of ax-ext 2602, so that ax-9 1999 will appear in lists of axioms used by a proof, since df-cleq 2615 implies ax-9 1999 by theorem bj-ax9 32890. We may revisit this in the future. (Revised by NM, 28-Oct-2021.) (Proof modification is discouraged.)
|- ( A = B <-> A. x ( x e. A <-> x e. B ) )
Theorem	cvjust 2617*	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 1482, which allows us to substitute a setvar variable for a class variable. See also cab 2608 and df-clab 2609. Note that this is not a rigorous justification, because cv 1482 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 2618*	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 2615 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2615 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1998), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2609. Alternate definitions of A e. B (but that require either A or B to be a set) are shown by clel2 3339, clel3 3341, and clel4 3342. 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. While the three class definitions df-clab 2609, df-cleq 2615, and df-clel 2618 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker. For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-1993.)
|- ( A e. B <-> E. x ( x = A /\ x e. B ) )
Theorem	eqriv 2619*	Infer equality of classes from equivalence of membership. (Contributed by NM, 21-Jun-1993.)
|- ( x e. A <-> x e. B )   =>    |- A = B
Theorem	eqrdv 2620*	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 2621*	Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
|- ( ( 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 2622	Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41. This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also biid 251. An early mention of this law can be found in Aristotle, Metaphysics, Z.17, 1041a10-20. (Thanks to Stefan Allan and BJ for this information.) (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Oct-2017.)
|- A = A
Theorem	eqidd 2623	Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
|- ( ph -> A = A )
Theorem	eqeq1d 2624	Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A = C <-> B = C ) )
Theorem	eqeq1dALT 2625	Shorter proof of eqeq1d 2624 based on more axioms. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 19-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A = C <-> B = C ) )
Theorem	eqeq1 2626	Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
|- ( A = B -> ( A = C <-> B = C ) )
Theorem	eqeq1i 2627	Inference from equality to equivalence of equalities. (Contributed by NM, 15-Jul-1993.)
|- A = B   =>    |- ( A = C <-> B = C )
Theorem	eqcomd 2628	Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) Allow shortening of eqcom 2629. (Revised by Wolf Lammen, 19-Nov-2019.)
|- ( ph -> A = B )   =>    |- ( ph -> B = A )
Theorem	eqcom 2629	Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
|- ( A = B <-> B = A )
Theorem	eqcoms 2630	Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 24-Jun-1993.)
|- ( A = B -> ph )   =>    |- ( B = A -> ph )
Theorem	eqcomi 2631	Inference from commutative law for class equality. (Contributed by NM, 26-May-1993.)
|- A = B   =>    |- B = A
Theorem	eqeq2d 2632	Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Allow shortening of eqeq2 2633. (Revised by Wolf Lammen, 19-Nov-2019.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C = A <-> C = B ) )
Theorem	eqeq2 2633	Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
|- ( A = B -> ( C = A <-> C = B ) )
Theorem	eqeq2i 2634	Inference from equality to equivalence of equalities. (Contributed by NM, 26-May-1993.)
|- A = B   =>    |- ( C = A <-> C = B )
Theorem	eqeq12 2635	Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
|- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )
Theorem	eqeq12i 2636	A useful inference for substituting definitions into an equality. (Contributed by NM, 15-Jul-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
|- A = B   &    |- C = D   =>    |- ( A = C <-> B = D )
Theorem	eqeq12d 2637	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 2638	A useful inference for substituting definitions into an equality. See also eqeqan12dALT 2639. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )
Theorem	eqeqan12dALT 2639	Alternate proof of eqeqan12d 2638. This proof has one more step but one fewer essential step. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )
Theorem	eqeqan12rd 2640	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 2641	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 2642	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 2643	A transitive law for class equality. (Contributed by NM, 20-May-2005.)
|- ( ( A = C /\ B = C ) -> A = B )
Theorem	eqtri 2644	An equality transitivity inference. (Contributed by NM, 26-May-1993.)
|- A = B   &    |- B = C   =>    |- A = C
Theorem	eqtr2i 2645	An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
|- A = B   &    |- B = C   =>    |- C = A
Theorem	eqtr3i 2646	An equality transitivity inference. (Contributed by NM, 6-May-1994.)
|- A = B   &    |- A = C   =>    |- B = C
Theorem	eqtr4i 2647	An equality transitivity inference. (Contributed by NM, 26-May-1993.)
|- A = B   &    |- C = B   =>    |- A = C
Theorem	3eqtri 2648	An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
|- A = B   &    |- B = C   &    |- C = D   =>    |- A = D
Theorem	3eqtrri 2649	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 2650	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
|- A = B   &    |- C = B   &    |- C = D   =>    |- A = D
Theorem	3eqtr2ri 2651	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 2652	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 2653	An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
|- A = B   &    |- A = C   &    |- B = D   =>    |- D = C
Theorem	3eqtr4i 2654	An inference from three chained equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &    |- C = A   &    |- D = B   =>    |- C = D
Theorem	3eqtr4ri 2655	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 2656	An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
|- ( ph -> A = B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A = C )
Theorem	eqtr2d 2657	An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
|- ( ph -> A = B )   &    |- ( ph -> B = C )   =>    |- ( ph -> C = A )
Theorem	eqtr3d 2658	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
|- ( ph -> A = B )   &    |- ( ph -> A = C )   =>    |- ( ph -> B = C )
Theorem	eqtr4d 2659	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
|- ( ph -> A = B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A = C )
Theorem	3eqtrd 2660	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 2661	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 2662	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 2663	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 2664	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 2665	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 2666	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 2667	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	syl5eq 2668	An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
|- A = B   &    |- ( ph -> B = C )   =>    |- ( ph -> A = C )
Theorem	syl5req 2669	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- A = B   &    |- ( ph -> B = C )   =>    |- ( ph -> C = A )
Theorem	syl5eqr 2670	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
|- B = A   &    |- ( ph -> B = C )   =>    |- ( ph -> A = C )
Theorem	syl5reqr 2671	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- B = A   &    |- ( ph -> B = C )   =>    |- ( ph -> C = A )
Theorem	syl6eq 2672	An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
|- ( ph -> A = B )   &    |- B = C   =>    |- ( ph -> A = C )
Theorem	syl6req 2673	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- ( ph -> A = B )   &    |- B = C   =>    |- ( ph -> C = A )
Theorem	syl6eqr 2674	An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
|- ( ph -> A = B )   &    |- C = B   =>    |- ( ph -> A = C )
Theorem	syl6reqr 2675	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- ( ph -> A = B )   &    |- C = B   =>    |- ( ph -> C = A )
Theorem	sylan9eq 2676	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 2677	An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
|- ( ph -> B = A )   &    |- ( ps -> B = C )   =>    |- ( ( ph /\ ps ) -> A = C )
Theorem	sylan9eqr 2678	An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
|- ( ph -> A = B )   &    |- ( ps -> B = C )   =>    |- ( ( ps /\ ph ) -> A = C )
Theorem	3eqtr3g 2679	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 2680	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 2681	A chained equality inference, useful for converting to definitions. (Contributed by NM, 21-Jun-1993.)
|- ( ph -> A = B )   &    |- C = A   &    |- D = B   =>    |- ( ph -> C = D )
Theorem	3eqtr4a 2682	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 2683	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 2684	Weaker version of eleq1 2689 (but more general than elequ1 1997) not depending on ax-ext 2602 (nor ax-12 2047 nor df-cleq 2615). (Contributed by BJ, 24-Jun-2019.)
|- ( x = y -> ( x e. A <-> y e. A ) )
Theorem	eleq2w 2685	Weaker version of eleq2 2690 (but more general than elequ2 2004) not depending on ax-ext 2602 (nor ax-12 2047 nor df-cleq 2615). (Contributed by BJ, 29-Sep-2019.)
|- ( x = y -> ( A e. x <-> A e. y ) )
Theorem	eleq1d 2686	Deduction from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993.) Allow shortening of eleq1 2689. (Revised by Wolf Lammen, 20-Nov-2019.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A e. C <-> B e. C ) )
Theorem	eleq2d 2687	Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C e. A <-> C e. B ) )
Theorem	eleq2dALT 2688	Alternate proof of eleq2d 2687, shorter at the expense of using more axioms. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 20-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C e. A <-> C e. B ) )
Theorem	eleq1 2689	Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
|- ( A = B -> ( A e. C <-> B e. C ) )
Theorem	eleq2 2690	Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
|- ( A = B -> ( C e. A <-> C e. B ) )
Theorem	eleq12 2691	Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
|- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) )
Theorem	eleq1i 2692	Inference from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993.)
|- A = B   =>    |- ( A e. C <-> B e. C )
Theorem	eleq2i 2693	Inference from equality to equivalence of membership. (Contributed by NM, 26-May-1993.)
|- A = B   =>    |- ( C e. A <-> C e. B )
Theorem	eleq12i 2694	Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
|- A = B   &    |- C = D   =>    |- ( A e. C <-> B e. D )
Theorem	eleq12d 2695	Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A e. C <-> B e. D ) )
Theorem	eleq1a 2696	A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
|- ( A e. B -> ( C = A -> C e. B ) )
Theorem	eqeltri 2697	Substitution of equal classes into membership relation. (Contributed by NM, 21-Jun-1993.)
|- A = B   &    |- B e. C   =>    |- A e. C
Theorem	eqeltrri 2698	Substitution of equal classes into membership relation. (Contributed by NM, 21-Jun-1993.)
|- A = B   &    |- A e. C   =>    |- B e. C
Theorem	eleqtri 2699	Substitution of equal classes into membership relation. (Contributed by NM, 15-Jul-1993.)
|- A e. B   &    |- B = C   =>    |- A e. C
Theorem	eleqtrri 2700	Substitution of equal classes into membership relation. (Contributed by NM, 15-Jul-1993.)
|- A e. B   &    |- C = B   =>    |- A e. C