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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 3340 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 3315 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 3315, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3314 in the form of sbc8g 3321. 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 3314 and assert that is always false when is a proper class. The theorem sbc2or 3322 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3315. The related definition dfcsb 3421 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 3313  . 2 
5  1, 2  cab 2428  . . 3 
6  3, 5  wcel 1804  . 2 
7  4, 6  wb 184  1 
Colors of variables: wff setvar class 
This definition is referenced by: dfsbcq 3315 dfsbcq2 3316 sbceqbid 3320 sbcex 3323 nfsbc1d 3331 nfsbcd 3334 cbvsbc 3342 sbcbi2 3364 sbcbid 3371 intab 4302 brab1 4482 iotacl 5564 riotasbc 6258 scottexs 8308 scott0s 8309 hta 8318 issubc 15186 dmdprd 17008 sbceqbidf 27358 setinds 29186 bnj1454 33768 bnj110 33784 bjcsbsnlem 34353 frege54cor1c 37746 frege55lem1c 37747 frege55c 37749 
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