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Mirrors > Home > MPE Home > Th. List > dfsbc  Unicode version 
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 dfsbc 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 dfsbc 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 dfcsb 3314 defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14Apr1995.) (Revised by NM, 25Dec2016.) 
Ref  Expression 

dfsbc 
Step  Hyp  Ref  Expression 

1  wph  . . 3  
2  vx  . . 3  
3  cA  . . 3  
4  1, 2, 3  wsbc 3212  . 2 
5  1, 2  cab 2467  . . 3 
6  3, 5  wcel 1724  . 2 
7  4, 6  wb 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 bjcsbsnlem 30665 
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