Definition df-sbc 3045

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 3070 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 3046 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 3046, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3045 in the form of sbc8g 3052. 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 3045 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 3044	. 2 wff [. A / x ]. ph
5	1, 2	cab 2220	. . 3 class { x | ph }
6	3, 5	wcel 2205	. 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  3046  dfsbcq2  3047  sbceqbid  3051  sbcex  3053  nfsbc1d  3061  nfsbcd  3064  cbvsbcw  3072  cbvsbc  3073  sbcbi2  3095  sbcbid  3102  csbcow  3151  nfsbcdw  3174  intab  3980  brab1  4159  iotacl  5339  riotasbc  6022  bdsbcALT  16646