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Definition df-sbc 3213
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 3238 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 3214 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 3214, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3213 in the form of sbc8g 3219. 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 3213 and assert that is always false when is a proper class.

The theorem sbc2or 3220 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3214.

The related definition df-csb 3314 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 3212 . 2
51, 2cab 2467 . . 3
63, 5wcel 1724 . 2
74, 6wb 178 1
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  3214  dfsbcq2  3215  sbcex  3221  nfsbc1d  3229  nfsbcd  3232  cbvsbc  3240  sbcbi2  3262  sbcbid  3269  intab  4168  brab1  4347  iotacl  5403  riotasbc  6043  scottexs  7925  scott0s  7926  hta  7935  issubc  14526  dmdprd  16062  sbceqbid  24481  sbceqbidf  24482  setinds  26235  bnj1454  30406  bnj110  30422  bj-csbsnlem  30665
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