|Description: Define the proper
substitution of a class for a set.
When A is a
proper class, our definition evaluates to false. This is
somewhat arbitrary: we could have, instead, chosen the conclusion of
sbc6 2783 for our definition, which always evaluates to
true for proper
Our definition also does not produce the same results as discussed in the
proof of Theorem 6.6 of [Quine] p. 42
(although Theorem 6.6 itself does
hold, as shown by dfsbcq 2760 below). Unfortunately, Quine's definition
requires a recursive syntactical breakdown of φ, and it does not
seem possible to express it with a single closed formula.
If we did not want to commit to any specific proper class behavior, we
could use this definition only to prove theorem dfsbcq 2760, which holds
for both our definition and Quine's, and from which we can derive a weaker
version of df-sbc 2759 in the form of sbc8g 2765. However, the behavior of
Quine's definition at proper classes is similarly arbitrary, and for
practical reasons (to avoid having to prove sethood of A in every use
of this definition) we allow direct reference to df-sbc 2759 and assert that
[A / x]φ is always false when A is a proper class.
The related definition df-csb defines proper substitution into a class
variable (as opposed to a wff variable). (Contributed by NM,
14-Apr-1995.) (Revised by NM, 25-Dec-2016.)