Definition df-sbc 2999

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 3024 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 3000 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 3000, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2999 in the form of sbc8g 3006. 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 2999 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 2998	. 2 wff [. A / x ]. ph
5	1, 2	cab 2191	. . 3 class { x | ph }
6	3, 5	wcel 2176	. 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  3000  dfsbcq2  3001  sbceqbid  3005  sbcex  3007  nfsbc1d  3015  nfsbcd  3018  cbvsbcw  3026  cbvsbc  3027  sbcbi2  3049  sbcbid  3056  csbcow  3104  nfsbcdw  3127  intab  3914  brab1  4091  iotacl  5256  riotasbc  5915  bdsbcALT  15795