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 2393 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 zfauscl 4143 to inex1 4155 (look at the instance of
zfauscl 4143 that occurs in the proof of inex1 4155). 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. An example is
cp 7561, which derives a formula containing wff
variables from
substitution instances of the class variables in its equivalent
formulation cplem2 7560. 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.) |