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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 3269 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 3244 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 syntactic 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 3244, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3243 in the form of sbc8g 3250. 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 3243 and assert that is always false when is a proper class. The theorem sbc2or 3251 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3244. The related definition dfcsb 3339 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 3242  . 2 
5  1, 2  cab 2414  . . 3 
6  3, 5  wcel 1872  . 2 
7  4, 6  wb 187  1 
Colors of variables: wff setvar class 
This definition is referenced by: dfsbcq 3244 dfsbcq2 3245 sbceqbid 3249 sbcex 3252 nfsbc1d 3260 nfsbcd 3263 cbvsbc 3271 sbcbi2 3291 sbcbid 3296 intab 4229 brab1 4412 iotacl 5531 riotasbc 6226 scottexs 8310 scott0s 8311 hta 8320 issubc 15683 dmdprd 17573 sbceqbidf 28059 bnj1454 29605 bnj110 29621 setinds 30375 bjcsbsnlem 31416 frege54cor1c 36424 frege55lem1c 36425 frege55c 36427 
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