| Description: 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 abbib 2356 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 
(that has a free variable  ) to a theorem with a class variable
 , we substitute
 for throughout and simplify,
where is a new
class variable not already in the wff. An example
is the conversion of sepgi 4250 to inex1 4265 (look at the instance of sepgi 4250
that occurs in the proof of inex1 4265). Conversely, to convert a theorem
with a class variable to one with , we substitute
   for
throughout and
simplify, where and
are new setvar and wff variables not already in the wff. For more
information on class variables, see Quine pp. 15-21 and/or Takeuti and
Zaring pp. 10-13.
(Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen,
12-Feb-2025.) |
