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 abbi2588 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 zfauscl4575 to inex14593 (look at the instance of
zfauscl4575 that occurs in the proof of inex14593). 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. An example is
cp8330, which derives a formula containing wff
variables from
substitution instances of the class variables in its equivalent
formulation cplem28329. 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.)