Definition df-sbc 2990

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 3015 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 2991 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 2991, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2990 in the form of sbc8g 2997. 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 2990 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 2989	. 2 wff [. A / x ]. ph
5	1, 2	cab 2182	. . 3 class { x | ph }
6	3, 5	wcel 2167	. 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  2991  dfsbcq2  2992  sbceqbid  2996  sbcex  2998  nfsbc1d  3006  nfsbcd  3009  cbvsbcw  3017  cbvsbc  3018  sbcbi2  3040  sbcbid  3047  csbcow  3095  nfsbcdw  3118  intab  3903  brab1  4080  iotacl  5243  riotasbc  5893  bdsbcALT  15505