Description: Define the proper substitution of a class for a set.

When A is a proper class, our definition evaluates to false (see
sbcex ). This is somewhat arbitrary: we could have, instead, chosen the
conclusion of sbc6 for our definition, whose right-hand side always
evaluates to true for proper classes.

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 below). For example, if A is a proper
class, Quine's substitution of A for y in 0 e. y evaluates to
0 e. A rather than our falsehood. (This can be seen by substituting
A , y , and 0 for alpha, beta, and gamma in Subcase 1 of
Quine's discussion on p. 42.) Unfortunately, Quine's definition requires
a recursive syntactic breakdown of ph , 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 definitiononly to prove theorem dfsbcq , which holds
for both our definition and Quine's, and from which we can derive a weaker
version of df-sbc in the form of sbc8g . 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 and assert that
[. A / x ]. ph is always false when A is a proper class.

The theorem sbc2or shows the apparently "strongest" statement we can
make regarding behavior at proper classes if we start from dfsbcq .

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)