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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 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 dfsbc 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 dfsbc 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 dfcsb 3266 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 3164  . 2 
5  1, 2  cab 2408  . . 3 
6  3, 5  wcel 1749  . 2 
7  4, 6  wb 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 bjcsbsnlem 31867 
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