mmtheorems22 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 2101-2200 - Page 22 of 191

Type	Label	Description
Statement
Theorem	moexex 2101	"At most one" double quantification.
|- (ph -> A.yph)   =>   |- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))
Theorem	moexexv 2102	"At most one" double quantification.
|- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))
Theorem	2moex 2103	Double quantification with "at most one."
|- (E*xE.yph -> A.yE*xph)
Theorem	2euex 2104	Double quantification with existential uniqueness. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)
|- (E!xE.yph -> E.yE!xph)
Theorem	2eumo 2105	Double quantification with existential uniqueness and "at most one."
|- (E!xE*yph -> E*xE!yph)
Theorem	2eu2ex 2106	Double existential uniqueness.
|- (E!xE!yph -> E.xE.yph)
Theorem	2moswap 2107	A condition allowing swap of "at most one" and existential quantifiers.
|- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
Theorem	2euswap 2108	A condition allowing swap of uniqueness and existential quantifiers.
|- (A.xE*yph -> (E!xE.yph -> E!yE.xph))
Theorem	2exeu 2109	Double existential uniqueness implies double uniqueness quantification.
|- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
Theorem	2mo 2110	Two equivalent expressions for double "at most one."
|- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
Theorem	2mos 2111	Double "exists at most one", using implicit substitition.
|- ((x = z /\ y = w) -> (ph <-> ps))   =>   |- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ ps) -> (x = z /\ y = w)))
Theorem	2eu1 2112	Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one.
|- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
Theorem	2eu2 2113	Double existential uniqueness.
|- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))
Theorem	2eu3 2114	Double existential uniqueness.
|- (A.xA.y(E*xph \/ E*yph) -> ((E!xE!yph /\ E!yE!xph) <-> (E!xE.yph /\ E!yE.xph)))
Theorem	2eu4 2115	This theorem provides us with a definition of double existential uniqueness ("exactly one x and exactly one y"). Naively one might think (incorrectly) that it could be defined by E!xE!yph. See 2eu1 2112 for a condition under which the naive definition holds and 2exeu 2109 for a one-way implication. See 2eu5 2116 and 2eu8 2119 for alternate definitions.
|- ((E!xE.yph /\ E!yE.xph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
Theorem	2eu5 2116	An alternate definition of double existential uniqueness (see 2eu4 2115). A mistake sometimes made in the literature is to use E!xE!y to mean "exactly one x and exactly one y." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining A.xE*yph as an additional condition. The correct definition apparently has never been published. (E* means "exists at most one.")
|- ((E!xE!yph /\ A.xE*yph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
Theorem	2eu6 2117	Two equivalent expressions for double existential uniqueness.
|- ((E!xE.yph /\ E!yE.xph) <-> E.zE.wA.xA.y(ph <-> (x = z /\ y = w)))
Theorem	2eu7 2118	Two equivalent expressions for double existential uniqueness.
|- ((E!xE.yph /\ E!yE.xph) <-> E!xE!y(E.xph /\ E.yph))
Theorem	2eu8 2119	Two equivalent expressions for double existential uniqueness. Curiously, we can put E! on either of the internal conjuncts but not both. We can also commute E!xE!y using 2eu7 2118.
|- (E!xE!y(E.xph /\ E.yph) <-> E!xE!y(E!xph /\ E.yph))
Theorem	euequ1 2120	Equality has existential uniqueness. Special case of eueq1 2673 proved using only predicate calculus. (Contributed by Stefan Allan, 4-Dec-2008.)
|- E!x x = y
Theorem	exists1 2121	Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 3662.
|- (E!x x = x <-> A.x x = y)
Theorem	exists2 2122	A condition implying that at least two things exist. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)
|- ((E.xph /\ E.x -. ph) -> -. E!x x = x)
ZF Set Theory - start with the Axiom of Extensionality
Introduce the Axiom of Extensionality
Axiom	ax-ext 2123	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 logical axioms, we would rewrite ax-8 1594 through ax-16 1854 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. 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 2123 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 3596, which involves a wff metavariable). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the the infinite axioms generated by the ax-ext 2123 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version.
|- (A.z(z e. x <-> z e. y) -> x = y)
Theorem	axext2 2124	The Axiom of Extensionality (ax-ext 2123) 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.
|- E.z((z e. x <-> z e. y) -> x = y)
Theorem	axext3 2125	A generalization of the Axiom of Extensionality in which x and y need not be distinct. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
|- (A.z(z e. x <-> z e. y) -> x = y)
Theorem	axext4 2126	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 2123 and df-cleq 2134.
|- (x = y <-> A.z(z e. x <-> z e. y))
Theorem	bm1.1 2127	Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462.
|- (ph -> A.xph)   =>   |- (E.xA.y(y e. x <-> ph) -> E!xA.y(y e. x <-> ph))
Class abstractions (a.k.a. class builders)
Syntax	cab 2128	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 3083). Note that a set variable can be expressed as a class builder per theorem cvjust 2136, justifying the assignment of set variables to class variables via the use of cv 1585.
class {x | ph}
Definition	df-clab 2129	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 1588, which extends or "overloads" the wel 1589 definition connecting set variables, requires that both sides of e. be a class. In df-cleq 2134 and df-clel 2137, 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 set variable. Syntax definition cv 1585 allows us to substitute a set variable x for a class variable: all sets are classes by cvjust 2136 (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 2248 for a quick overview). Because class variables can be substituted with compound expressions and set variables cannot, it is often useful to convert a theorem containing a free set variable to a more general version with a class variable. This is done with theorems such as vtoclg 2588 which is used, for example, to convert elirrv 5933 to elirr 5934.
|- (x e. {y | ph} <-> [x / y]ph)
Theorem	abid 2130	Simplification of class abstraction notation when the free and bound variables are identical.
|- (x e. {x | ph} <-> ph)
Theorem	hbab1 2131	Bound-variable hypothesis builder for a class abstraction.
|- (y e. {x | ph} -> A.x y e. {x | ph})
Theorem	hbab 2132	Bound-variable hypothesis builder for a class abstraction.
|- (ph -> A.xph)   =>   |- (z e. {y | ph} -> A.x z e. {y | ph})
Theorem	hbabd 2133	Deduction form of bound-variable hypothesis builder hbab 2132.
|- (ph -> A.xA.yph)   &   |- (ph -> (ps -> A.xps))   =>   |- (ph -> (z e. {y | ps} -> A.x z e. {y | ps}))
Definition	df-cleq 2134	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 2126). 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 2129, df-clel 2137, and abeq2 2248.
|- (A.x(x e. y <-> x e. z) -> y = z)   =>   |- (A = B <-> A.x(x e. A <-> x e. B))
Theorem	dfcleq 2135	The same as df-cleq 2134 with the hypothesis removed using the Axiom of Extensionality ax-ext 2123.
|- (A = B <-> A.x(x e. A <-> x e. B))
Theorem	cvjust 2136	Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a set variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1585, which allows us to substitute a set variable for a class variable. See also cab 2128 and df-clab 2129. Note that this is not a rigorous justification, because cv 1585 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."
|- x = {y | y e. x}
Definition	df-clel 2137	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 2134 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2134 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 1979), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2129. Alternate definitions of A e. B (but that require either A or B to be a set) are shown by clel2 2636, clel3 2638, and clel4 2639.
|- (A e. B <-> E.x(x = A /\ x e. B))
Theorem	eqriv 2138	Infer equality of classes from equivalence of membership.
|- (x e. A <-> x e. B)   =>   |- A = B
Theorem	eqrdv 2139	Deduce equality of classes from equivalence of membership.
|- (ph -> (x e. A <-> x e. B))   =>   |- (ph -> A = B)
Theorem	eqrdav 2140	Deduce equality of classes from an equivalence of membership that depends on the membership variable.
|- ((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 2141	Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41. This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). (Thanks to Stefan Allan for this information.)
|- A = A
Theorem	eqidd 2142	Class identity law with antecedent.
|- (ph -> A = A)
Theorem	eqcom 2143	Commutative law for class equality. Theorem 6.5 of [Quine] p. 41.
|- (A = B <-> B = A)
Theorem	eqcoms 2144	Inference applying commutative law for class equality to an antecedent.
|- (A = B -> ph)   =>   |- (B = A -> ph)
Theorem	eqcomi 2145	Inference from commutative law for class equality.
|- A = B   =>   |- B = A
Theorem	eqcomd 2146	Deduction from commutative law for class equality.
|- (ph -> A = B)   =>   |- (ph -> B = A)
Theorem	eqeq1 2147	Equality implies equivalence of equalities.
|- (A = B -> (A = C <-> B = C))
Theorem	eqeq1i 2148	Inference from equality to equivalence of equalities.
|- A = B   =>   |- (A = C <-> B = C)
Theorem	eqeq1d 2149	Deduction from equality to equivalence of equalities.
|- (ph -> A = B)   =>   |- (ph -> (A = C <-> B = C))
Theorem	eqeq2 2150	Equality implies equivalence of equalities.
|- (A = B -> (C = A <-> C = B))
Theorem	eqeq2i 2151	Inference from equality to equivalence of equalities.
|- A = B   =>   |- (C = A <-> C = B)
Theorem	eqeq2d 2152	Deduction from equality to equivalence of equalities.
|- (ph -> A = B)   =>   |- (ph -> (C = A <-> C = B))
Theorem	eqeq12 2153	Equality relationship among 4 classes.
|- ((A = B /\ C = D) -> (A = C <-> B = D))
Theorem	eqeq12i 2154	A useful inference for substituting definitions into an equality. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- C = D   =>   |- (A = C <-> B = D)
Theorem	eqeq12d 2155	A useful inference for substituting definitions into an equality. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A = C <-> B = D))
Theorem	eqeqan12d 2156	A useful inference for substituting definitions into an equality. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ph /\ ps) -> (A = C <-> B = D))
Theorem	eqeqan12rd 2157	A useful inference for substituting definitions into an equality.
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ps /\ ph) -> (A = C <-> B = D))
Theorem	eqtr 2158	Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13.
|- ((A = B /\ B = C) -> A = C)
Theorem	eqtr2 2159	A transitive law for class equality. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- ((A = B /\ A = C) -> B = C)
Theorem	eqtr3 2160	A transitive law for class equality.
|- ((A = C /\ B = C) -> A = B)
Theorem	eqtri 2161	An equality transitivity inference.
|- A = B   &   |- B = C   =>   |- A = C
Theorem	eqtr2i 2162	An equality transitivity inference.
|- A = B   &   |- B = C   =>   |- C = A
Theorem	eqtr3i 2163	An equality transitivity inference.
|- A = B   &   |- A = C   =>   |- B = C
Theorem	eqtr4i 2164	An equality transitivity inference.
|- A = B   &   |- C = B   =>   |- A = C
Theorem	3eqtri 2165	An inference from three chained equalities.
|- A = B   &   |- B = C   &   |- C = D   =>   |- A = D
Theorem	3eqtrri 2166	An inference from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- B = C   &   |- C = D   =>   |- D = A
Theorem	3eqtr2i 2167	An inference from three chained equalities.
|- A = B   &   |- C = B   &   |- C = D   =>   |- A = D
Theorem	3eqtr2ri 2168	An inference from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- C = B   &   |- C = D   =>   |- D = A
Theorem	3eqtr3i 2169	An inference from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- A = C   &   |- B = D   =>   |- C = D
Theorem	3eqtr3ri 2170	An inference from three chained equalities.
|- A = B   &   |- A = C   &   |- B = D   =>   |- D = C
Theorem	3eqtr4i 2171	An inference from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- C = A   &   |- D = B   =>   |- C = D
Theorem	3eqtr4ri 2172	An inference from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &   |- C = A   &   |- D = B   =>   |- D = C
Theorem	eqtrd 2173	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ph -> B = C)   =>   |- (ph -> A = C)
Theorem	eqtr2d 2174	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ph -> B = C)   =>   |- (ph -> C = A)
Theorem	eqtr3d 2175	An equality transitivity equality deduction.
|- (ph -> A = B)   &   |- (ph -> A = C)   =>   |- (ph -> B = C)
Theorem	eqtr4d 2176	An equality transitivity equality deduction.
|- (ph -> A = B)   &   |- (ph -> C = B)   =>   |- (ph -> A = C)
Theorem	3eqtrd 2177	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> B = C)   &   |- (ph -> C = D)   =>   |- (ph -> A = D)
Theorem	3eqtrrd 2178	A deduction from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ph -> B = C)   &   |- (ph -> C = D)   =>   |- (ph -> D = A)
Theorem	3eqtr2d 2179	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = B)   &   |- (ph -> C = D)   =>   |- (ph -> A = D)
Theorem	3eqtr2rd 2180	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = B)   &   |- (ph -> C = D)   =>   |- (ph -> D = A)
Theorem	3eqtr3d 2181	A deduction from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> C = D)
Theorem	3eqtr3rd 2182	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> D = C)
Theorem	3eqtr4d 2183	A deduction from three chained equalities. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> C = D)
Theorem	3eqtr4rd 2184	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> D = C)
Theorem	syl5eq 2185	An equality transitivity deduction.
|- A = B   &   |- (ph -> B = C)   =>   |- (ph -> A = C)
Theorem	syl5eqOLD 2186	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = A   =>   |- (ph -> C = B)
Theorem	syl5req 2187	An equality transitivity deduction.
|- A = B   &   |- (ph -> B = C)   =>   |- (ph -> C = A)
Theorem	syl5reqOLD 2188	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = A   =>   |- (ph -> B = C)
Theorem	syl5eqr 2189	An equality transitivity deduction.
|- B = A   &   |- (ph -> B = C)   =>   |- (ph -> A = C)
Theorem	syl5eqrOLD 2190	An equality transitivity deduction.
|- (ph -> A = B)   &   |- A = C   =>   |- (ph -> C = B)
Theorem	syl5reqr 2191	An equality transitivity deduction.
|- B = A   &   |- (ph -> B = C)   =>   |- (ph -> C = A)
Theorem	syl5reqrOLD 2192	An equality transitivity deduction.
|- (ph -> A = B)   &   |- A = C   =>   |- (ph -> B = C)
Theorem	syl6eq 2193	An equality transitivity deduction.
|- (ph -> A = B)   &   |- B = C   =>   |- (ph -> A = C)
Theorem	syl6req 2194	An equality transitivity deduction.
|- (ph -> A = B)   &   |- B = C   =>   |- (ph -> C = A)
Theorem	syl6eqr 2195	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = B   =>   |- (ph -> A = C)
Theorem	syl6reqr 2196	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = B   =>   |- (ph -> C = A)
Theorem	sylan9eq 2197	An equality transitivity deduction. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A = B)   &   |- (ps -> B = C)   =>   |- ((ph /\ ps) -> A = C)
Theorem	sylan9req 2198	An equality transitivity deduction.
|- (ph -> B = A)   &   |- (ps -> B = C)   =>   |- ((ph /\ ps) -> A = C)
Theorem	sylan9eqr 2199	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ps -> B = C)   =>   |- ((ps /\ ph) -> A = C)
Theorem	3eqtr3g 2200	A chained equality inference, useful for converting from definitions.
|- (ph -> A = B)   &   |- A = C   &   |- B = D   =>   |- (ph -> C = D)