mmtheorems16 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1501-1600 - Page 16 of 105

Type	Label	Description
Statement
Theorem	syl6eqr 1501	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = B   =>   |- (ph -> A = C)
Theorem	syl6reqr 1502	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = B   =>   |- (ph -> C = A)
Theorem	sylan9eq 1503	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ps -> B = C)   =>   |- ((ph /\ ps) -> A = C)
Theorem	sylan9req 1504	An equality transitivity deduction.
|- (ph -> B = A)   &   |- (ps -> B = C)   =>   |- ((ph /\ ps) -> A = C)
Theorem	sylan9eqr 1505	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ps -> B = C)   =>   |- ((ps /\ ph) -> A = C)
Theorem	3eqtr3g 1506	A chained equality inference, useful for converting from definitions.
|- (ph -> A = B)   &   |- A = C   &   |- B = D   =>   |- (ph -> C = D)
Theorem	3eqtr4g 1507	A chained equality inference, useful for converting to definitions.
|- (ph -> A = B)   &   |- C = A   &   |- D = B   =>   |- (ph -> C = D)
Theorem	3eqtr4a 1508	A chained equality inference, useful for converting to definitions.
|- A = B   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> C = D)
Theorem	eq2tr 1509	A compound transitive inference for class equality.
|- (A = C -> D = F)   &   |- (B = D -> C = G)   =>   |- ((A = C /\ B = F) <-> (B = D /\ A = G))
Theorem	eleq1 1510	Equality implies equivalence of membership.
|- (A = B -> (A e. C <-> B e. C))
Theorem	eleq2 1511	Equality implies equivalence of membership.
|- (A = B -> (C e. A <-> C e. B))
Theorem	eleq12 1512	Equality implies equivalence of membership.
|- ((A = B /\ C = D) -> (A e. C <-> B e. D))
Theorem	eleq1i 1513	Inference from equality to equivalence of membership.
|- A = B   =>   |- (A e. C <-> B e. C)
Theorem	eleq2i 1514	Inference from equality to equivalence of membership.
|- A = B   =>   |- (C e. A <-> C e. B)
Theorem	eleq12i 1515	Inference from equality to equivalence of membership.
|- A = B   &   |- C = D   =>   |- (A e. C <-> B e. D)
Theorem	eleq1d 1516	Deduction from equality to equivalence of membership.
|- (ph -> A = B)   =>   |- (ph -> (A e. C <-> B e. C))
Theorem	eleq2d 1517	Deduction from equality to equivalence of membership.
|- (ph -> A = B)   =>   |- (ph -> (C e. A <-> C e. B))
Theorem	eleq12d 1518	Deduction from equality to equivalence of membership.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A e. C <-> B e. D))
Theorem	eleq1a 1519	A transitive-type law relating membership and equality.
|- (A e. B -> (C = A -> C e. B))
Theorem	eqeltr 1520	Substitution of equal classes into membership relation.
|- A = B   &   |- B e. C   =>   |- A e. C
Theorem	eqeltrr 1521	Substitution of equal classes into membership relation.
|- A = B   &   |- A e. C   =>   |- B e. C
Theorem	eleqtr 1522	Substitution of equal classes into membership relation.
|- A e. B   &   |- B = C   =>   |- A e. C
Theorem	eleqtrr 1523	Substitution of equal classes into membership relation.
|- A e. B   &   |- C = B   =>   |- A e. C
Theorem	eqeltrd 1524	Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
|- (ph -> A = B)   &   |- (ph -> B e. C)   =>   |- (ph -> A e. C)
Theorem	eqeltrrd 1525	Deduction that substitutes equal classes into membership.
|- (ph -> A = B)   &   |- (ph -> A e. C)   =>   |- (ph -> B e. C)
Theorem	eleqtrd 1526	Deduction that substitutes equal classes into membership.
|- (ph -> A e. B)   &   |- (ph -> B = C)   =>   |- (ph -> A e. C)
Theorem	eleqtrrd 1527	Deduction that substitutes equal classes into membership.
|- (ph -> A e. B)   &   |- (ph -> C = B)   =>   |- (ph -> A e. C)
Theorem	syl5eqel 1528	A membership and equality inference.
|- (ph -> A e. B)   &   |- C = A   =>   |- (ph -> C e. B)
Theorem	syl5eqelr 1529	A membership and equality inference.
|- (ph -> A e. B)   &   |- A = C   =>   |- (ph -> C e. B)
Theorem	syl5eleq 1530	A membership and equality inference.
|- (ph -> A = B)   &   |- C e. A   =>   |- (ph -> C e. B)
Theorem	syl5eleqr 1531	A membership and equality inference.
|- (ph -> B = A)   &   |- C e. A   =>   |- (ph -> C e. B)
Theorem	syl6eqel 1532	A membership and equality inference.
|- (ph -> A = B)   &   |- B e. C   =>   |- (ph -> A e. C)
Theorem	syl6eqelr 1533	A membership and equality inference.
|- (ph -> B = A)   &   |- B e. C   =>   |- (ph -> A e. C)
Theorem	syl6eleq 1534	A membership and equality inference.
|- (ph -> A e. B)   &   |- B = C   =>   |- (ph -> A e. C)
Theorem	syl6eleqr 1535	A membership and equality inference.
|- (ph -> A e. B)   &   |- C = B   =>   |- (ph -> A e. C)
Theorem	cleqf 1536	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (A = B <-> A.x(x e. A <-> x e. B))
Theorem	nelneq 1537	A way of showing two classes are not equal.
|- ((A e. C /\ -. B e. C) -> -. A = B)
Theorem	nelneq2 1538	A way of showing two classes are not equal.
|- ((A e. B /\ -. A e. C) -> -. B = C)
Theorem	hbxfr 1539	A utility lemma to transfer a bound-variable hypothesis builder into a definition.
|- A = B   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. A -> A.x y e. A)
Theorem	hblem 1540	Lemma for hbeq 1541 and hbel 1542.
Theorem	hbeq 1541	If x is effectively bound in A and B, it is effectively bound in A = B.
|- (y e. A -> A.x y e. A)   &   |- (z e. B -> A.x z e. B)   =>   |- (A = B -> A.x A = B)
Theorem	hbel 1542	If x is effectively bound in A and B, it is effectively bound in A e. B.
|- (y e. A -> A.x y e. A)   &   |- (z e. B -> A.x z e. B)   =>   |- (A e. B -> A.x A e. B)
Theorem	hbeleq 1543	If x is effectively bound in y e. A, then it is effectively bound in y = A.
|- (y e. A -> A.x y e. A)   =>   |- (y = A -> A.x y = A)
Theorem	abeq2 1544	Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that eq2ab 1549 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable. Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable ph (that has a free variable x) to a theorem with a class variable A, we substitute x e. A for ph throughout and simplify, where A is a new class variable not already in the wff. An example is the conversion of zfauscl 2673 to inex1 2684 (look at the instance of zfauscl 2673 that occurs in the proof of inex1 2684). Conversely, to convert a theorem with a class variable A to one with ph, we substitute {x | ph} for A throughout and simplify, where x and ph are new set and wff variables not already in the wff. An example is cp 4646, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 4645.
|- (A = {x | ph} <-> A.x(x e. A <-> ph))
Theorem	abeq1 1545	Equality of a class variable and a class abstraction.
|- ({x | ph} = A <-> A.x(ph <-> x e. A))
Theorem	abeq2i 1546	Equality of a class variable and a class abstraction (inference rule).
|- A = {x | ph}   =>   |- (x e. A <-> ph)
Theorem	abeq1i 1547	Equality of a class variable and a class abstraction (inference rule).
|- {x | ph} = A   =>   |- (ph <-> x e. A)
Theorem	abeq2d 1548	Equality of a class variable and a class abstraction (deduction).
|- (ph -> A = {x | ps})   =>   |- (ph -> (x e. A <-> ps))
Theorem	eq2ab 1549	Equality of two class abstractions means their wff's are equivalent.
|- ({x | ph} = {x | ps} <-> A.x(ph <-> ps))
Theorem	abbi2i 1550	Equality of a class variable and a class abstraction (inference rule).
|- (x e. A <-> ph)   =>   |- A = {x | ph}
Theorem	abbii 1551	Equivalent wff's yield equal class abstractions (inference rule).
|- (ph <-> ps)   =>   |- {x | ph} = {x | ps}
Theorem	abbid 1552	Equivalent wff's yield equal class abstractions (deduction rule).
|- (ph -> A.xph)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> {x | ps} = {x | ch})
Theorem	abbidv 1553	Equivalent wff's yield equal class abstractions (deduction rule).
|- (ph -> (ps <-> ch))   =>   |- (ph -> {x | ps} = {x | ch})
Theorem	abbi2dv 1554	Deduction from a wff to a class abstraction.
|- (ph -> (x e. A <-> ps))   =>   |- (ph -> A = {x | ps})
Theorem	abbi1dv 1555	Deduction from a wff to a class abstraction.
|- (ph -> (ps <-> x e. A))   =>   |- (ph -> {x | ps} = A)
Theorem	abid2 1556	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35.
|- {x | x e. A} = A
Theorem	clelab 1557	Membership of a class variable in a class abstraction.
|- (A e. {x | ph} <-> E.x(x = A /\ ph))
Theorem	clabel 1558	Membership of a class abstraction in another class
|- ({x | ph} e. A <-> E.y(y e. A /\ A.x(x e. y <-> ph)))
Theorem	sbab 1559	The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable.
|- (x = y -> A = {z | [y / x]z e. A})
Theorem	sbabel 1560	Theorem to move a substitution in and out of a class abstraction.
|- (w e. A -> A.x w e. A)   =>   |- ([y / x]{z | ph} e. A <-> {z | [y / x]ph} e. A)
Negated equality and membership
Syntax	wne 1561	Extend wff notation to include inequality.
wff A =/= B
Syntax	wnel 1562	Extend wff notation to include negated membership.
wff A e/ B
Definition	df-ne 1563	Define inequality.
|- (A =/= B <-> -. A = B)
Definition	df-nel 1564	Define negated membership.
|- (A e/ B <-> -. A e. B)
Theorem