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 2542 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 4491 to inex1 4508 (look at the instance of
zfauscl 4491 that occurs in the proof of inex1 4508). 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. Examples
include dfsymdif2 3643 and cp 8314; the latter derives a formula
containing
wff variables from substitution instances of the class variables in its
equivalent formulation cplem2 8313. 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.) |