Definition df-sbc 2957

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 2981 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 2958 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 2958, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2957 in the form of sbc8g 2963. 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 2957 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 2956	. 2 wff [. A / x ]. ph
5	1, 2	cab 2157	. . 3 class { x | ph }
6	3, 5	wcel 2142	. 2 wff A e. { x | ph }
7	4, 6	wb 104	1 wff ( [. A / x ]. ph <-> A e. { x | ph } )

This definition is referenced by:  dfsbcq  2958  dfsbcq2  2959  sbcex  2964  nfsbc1d  2972  nfsbcd  2975  cbvsbcw  2983  cbvsbc  2984  sbcbi2  3006  sbcbid  3013  csbcow  3061  nfsbcdw  3084  intab  3861  brab1  4037  iotacl  5185  riotasbc  5828  bdsbcALT  13979