Definition df-sbc 3436

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 3462 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 3437 below). For example, if A is a proper class, Quine's substitution of A for y in 0 e. y evaluates to 0 e. A rather than our falsehood. (This can be seen by substituting A, y, and 0 for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactic 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 3437, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3436 in the form of sbc8g 3443. 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 3436 and assert that [. A / x ]. ph is always false when A is a proper class.

The theorem sbc2or 3444 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3437.

The related definition df-csb 3534 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 3435	. 2 wff [. A / x ]. ph
5	1, 2	cab 2608	. . 3 class { x | ph }
6	3, 5	wcel 1990	. 2 wff A e. { x | ph }
7	4, 6	wb 196	1 wff ( [. A / x ]. ph <-> A e. { x | ph } )

This definition is referenced by:  dfsbcq  3437  dfsbcq2  3438  sbceqbid  3442  sbcex  3445  nfsbc1d  3453  nfsbcd  3456  cbvsbc  3464  sbcbi2  3484  sbcbid  3489  intab  4507  brab1  4700  iotacl  5874  riotasbc  6626  scottexs  8750  scott0s  8751  hta  8760  issubc  16495  dmdprd  18397  sbceqbidf  29321  bnj1454  30912  bnj110  30928  setinds  31683  bj-csbsnlem  32898  frege54cor1c  38209  frege55lem1c  38210  frege55c  38212