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Mirrors > Home > MPE Home > Th. List > dfsbc  Structured version 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 3188 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 3164 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 3164, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3163 in the form of sbc8g 3169. 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 3163 and assert that is always false when is a proper class. The theorem sbc2or 3170 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3164. The related definition dfcsb 3253 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 3162  . 2 
5  1, 2  cab 2423  . . 3 
6  3, 5  wcel 1726  . 2 
7  4, 6  wb 178  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 3164 dfsbcq2 3165 sbcex 3171 nfsbc1d 3179 nfsbcd 3182 cbvsbc 3190 sbcbid 3215 intab 4081 brab1 4258 iotacl 5442 riotasbc 6566 scottexs 7812 scott0s 7813 hta 7822 issubc 14036 dmdprd 15560 setinds 25406 bnj1454 29214 bnj110 29230 
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