|Description: Define the proper
substitution of a class for a set.
When is a proper
class, our definition evaluates to false. This is
somewhat arbitrary: we could have, instead, chosen the conclusion of
sbc6 3340 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 3315 below). For example, if is a proper
class, Quine's substitution of for in
rather than our falsehood. (This can be seen by
, , and for alpha, beta, and gamma in Subcase 1
Quine's discussion on p. 42.) Unfortunately, Quine's definition requires
a recursive syntactical breakdown of , 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 3315, which holds
for both our definition and Quine's, and from which we can derive a weaker
version of df-sbc 3314 in the form of sbc8g 3321. However, the behavior of
Quine's definition at proper classes is similarly arbitrary, and for
practical reasons (to avoid having to prove sethood of in every use
of this definition) we allow direct reference to df-sbc 3314 and assert that
always false when is
a proper class.
The theorem sbc2or 3322 shows the apparently "strongest"
statement we can
make regarding behavior at proper classes if we start from dfsbcq 3315.
The related definition df-csb 3421 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.)