Definition df-sbc 2963

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 2988 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 2964 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 2964, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2963 in the form of sbc8g 2970. 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 2963 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 2962	. 2 wff [. A / x ]. ph
5	1, 2	cab 2163	. . 3 class { x | ph }
6	3, 5	wcel 2148	. 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  2964  dfsbcq2  2965  sbceqbid  2969  sbcex  2971  nfsbc1d  2979  nfsbcd  2982  cbvsbcw  2990  cbvsbc  2991  sbcbi2  3013  sbcbid  3020  csbcow  3068  nfsbcdw  3091  intab  3873  brab1  4050  iotacl  5201  riotasbc  5845  bdsbcALT  14547