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Definition df-sbc 3225
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 3250 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 3226 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 3226, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3225 in the form of sbc8g 3231. 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 3225 and assert that is always false when is a proper class.

The theorem sbc2or 3232 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3226.

The related definition df-csb 3326 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 3224 . 2
51, 2cab 2475 . . 3
63, 5wcel 1732 . 2
74, 6wb 178 1
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  3226  dfsbcq2  3227  sbcex  3233  nfsbc1d  3241  nfsbcd  3244  cbvsbc  3252  sbcbi2  3274  sbcbid  3281  intab  4184  brab1  4363  iotacl  5424  riotasbc  6079  scottexs  7980  scott0s  7981  hta  7990  issubc  14589  dmdprd  16174  sbceqbid  24993  sbceqbidf  24994  setinds  26744  bnj1454  30683  bnj110  30699  bj-csbsnlem  30942
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