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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 3279 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 3254 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 3254, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3253 in the form of sbc8g 3260. 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 3253 and assert that is always false when is a proper class. The theorem sbc2or 3261 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3254. The related definition dfcsb 3349 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 3252  . 2 
5  1, 2  cab 2367  . . 3 
6  3, 5  wcel 1826  . 2 
7  4, 6  wb 184  1 
Colors of variables: wff setvar class 
This definition is referenced by: dfsbcq 3254 dfsbcq2 3255 sbceqbid 3259 sbcex 3262 nfsbc1d 3270 nfsbcd 3273 cbvsbc 3281 sbcbi2 3301 sbcbid 3306 intab 4230 brab1 4412 iotacl 5483 riotasbc 6173 scottexs 8218 scott0s 8219 hta 8228 issubc 15241 dmdprd 17142 sbceqbidf 27497 setinds 29375 bnj1454 34247 bnj110 34263 bjcsbsnlem 34817 frege54cor1c 38414 frege55lem1c 38415 frege55c 38417 
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