Definition df-clab 1464

Description: 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 958, which extends or "overloads" the wel 959 definition connecting set variables, requires that both sides of e. be a class. In df-cleq 1469 and df-clel 1472, 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 955 allows us to substitute a set variable x for a class variable: all sets are classes by cvjust 1471 (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 1568 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 1847 which is used, for example, to convert elirrv 4598 to elirr 4599.

Assertion

Ref	Expression
df-clab	|- (x e. {y | ph} <-> [x / y]ph)

Detailed syntax breakdown of Definition df-clab

Step	Hyp	Ref	Expression
1		vx	. . . 4 set x
2	1	cv 955	. . 3 class x
3		wph	. . . 4 wff ph
4		vy	. . . 4 set y
5	3, 4	cab 1463	. . 3 class {y | ph}
6	2, 5	wcel 958	. 2 wff x e. {y | ph}
7	3, 4, 2	wsbc 1170	. 2 wff [x / y]ph
8	6, 7	wb 146	1 wff (x e. {y | ph} <-> [x / y]ph)

This definition is referenced by:  abid 1465  hbab1 1466  hbab 1467  hbabd 1468  cvjust 1471  clelab 1581  csbabg 2043  unab 2267  inab 2268  difab 2269  exss 2769  abrexex2 3871  scottexs 4718  scott0s 4719

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