| 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 abbi 2357 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. Conversely,
to convert a theorem with a class variable to one with , we
substitute    for throughout and simplify, where
and are
new set 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,
5-Aug-1993.) |
is distinct from all other
variables.