| 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 2814 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 𝜑
(that has a free variable 𝑥) to a theorem with a class variable
𝐴, we substitute 𝑥 ∈ 𝐴 for 𝜑
throughout and simplify,
where 𝐴 is a new class variable not already
in the wff. An example
is the conversion of zfauscl 5319 to inex1 5335 (look at the instance of
zfauscl 5319 that occurs in the proof of inex1 5335). Conversely, to convert
a theorem with a class variable 𝐴 to one with 𝜑, we substitute
{𝑥
∣ 𝜑} for 𝐴
throughout and simplify, where 𝑥 and 𝜑
are new setvar and wff variables not already in the wff. Examples
include dfsymdif2 4280 and cp 9960; the latter derives a formula
containing
wff variables from substitution instances of the class variables in its
equivalent formulation cplem2 9959. For more information on class
variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13.
Usage of eqabbw 2818 is preferred since it requires fewer axioms.
(Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen,
12-Feb-2025.) |
