Definition df-clab 2126

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 1480, which extends or "overloads" the wel 1481 definition connecting setvar variables, requires that both sides of e. be a class. In df-cleq 2132 and df-clel 2135, 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 1330 allows us to substitute a setvar variable x for a class variable: all sets are classes by cvjust 2134 (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 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 2248. (Contributed by NM, 5-Aug-1993.)

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 setvar x
2	1	cv 1330	. . 3 class x
3		wph	. . . 4 wff ph
4		vy	. . . 4 setvar y
5	3, 4	cab 2125	. . 3 class { y | ph }
6	2, 5	wcel 1480	. 2 wff x e. { y | ph }
7	3, 4, 1	wsb 1735	. 2 wff [ x / y ] ph
8	6, 7	wb 104	1 wff ( x e. { y | ph } <-> [ x / y ] ph )

This definition is referenced by:  abid  2127  hbab1  2128  hbab  2130  cvjust  2134  abbi  2253  sb8ab  2261  cbvabw  2262  cbvab  2263  clelab  2265  nfabd  2300  vjust  2687  dfsbcq2  2912  sbc8g  2916  csbabg  3061  unab  3343  inab  3344  difab  3345  rabeq0  3392  abeq0  3393  oprcl  3729  exss  4149  peano1  4508  peano2  4509  iotaeq  5096  nfvres  5454  abrexex2g  6018  opabex3d  6019  opabex3  6020  abrexex2  6022  bdab  13036  bdph  13048  bdcriota  13081