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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 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 dfsbc 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 dfsbc 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 dfcsb 3326 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 3224  . 2 
5  1, 2  cab 2475  . . 3 
6  3, 5  wcel 1732  . 2 
7  4, 6  wb 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 bjcsbsnlem 30942 
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