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Definition df-sbc 3165
Description: Define the proper substitution of a class for a set.

When is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3190 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 3166 below). For example, if is a proper class, Quine's substitution of for in evaluates to rather than our falsehood. (This can be seen by substituting , , and for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactical breakdown of , 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 3166, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3165 in the form of sbc8g 3171. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of in every use of this definition) we allow direct reference to df-sbc 3165 and assert that is always false when is a proper class.

The theorem sbc2or 3172 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3166.

The related definition df-csb 3266 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.)

Ref Expression

Detailed syntax breakdown of Definition df-sbc
StepHypRef Expression
1 wph . . 3
2 vx . . 3
3 cA . . 3
41, 2, 3wsbc 3164 . 2
51, 2cab 2408 . . 3
63, 5wcel 1749 . 2
74, 6wb 178 1
Colors of variables: wff setvar class
This definition is referenced by:  dfsbcq  3166  dfsbcq2  3167  sbcex  3173  nfsbc1d  3181  nfsbcd  3184  cbvsbc  3192  sbcbi2  3214  sbcbid  3221  intab  4133  brab1  4312  iotacl  5376  riotasbc  6037  scottexs  8041  scott0s  8042  hta  8051  issubc  14688  dmdprd  16368  sbceqbid  25544  sbceqbidf  25545  setinds  27293  bnj1454  31413  bnj110  31429  bj-csbsnlem  31867
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