Definition df-sbc 2998

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 3023 for our definition, which 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 2999 below). Unfortunately, Quine's definition requires a recursive syntactical 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 definition only to prove Theorem dfsbcq 2999, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2998 in the form of sbc8g 3005. 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 2998 and assert that [. A / x ]. ph 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.)

Assertion

Ref	Expression
df-sbc	|- ( [. A / x ]. ph <-> A e. { x | ph } )

Detailed syntax breakdown of Definition df-sbc

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		vx	. . 3 setvar x
3		cA	. . 3 class A
4	1, 2, 3	wsbc 2997	. 2 wff [. A / x ]. ph
5	1, 2	cab 2190	. . 3 class { x | ph }
6	3, 5	wcel 2175	. 2 wff A e. { x | ph }
7	4, 6	wb 105	1 wff ( [. A / x ]. ph <-> A e. { x | ph } )

This definition is referenced by:  dfsbcq  2999  dfsbcq2  3000  sbceqbid  3004  sbcex  3006  nfsbc1d  3014  nfsbcd  3017  cbvsbcw  3025  cbvsbc  3026  sbcbi2  3048  sbcbid  3055  csbcow  3103  nfsbcdw  3126  intab  3913  brab1  4090  iotacl  5255  riotasbc  5914  bdsbcALT  15757