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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 3147 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 3123 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 3123, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3122 in the form of sbc8g 3128. 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 3122 and assert that is always false when is a proper class. The theorem sbc2or 3129 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3123. The related definition dfcsb 3212 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 3121  . 2 
5  1, 2  cab 2390  . . 3 
6  3, 5  wcel 1721  . 2 
7  4, 6  wb 177  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 3123 dfsbcq2 3124 sbcex 3130 nfsbc1d 3138 nfsbcd 3141 cbvsbc 3149 sbcbid 3174 intab 4040 brab1 4217 iotacl 5400 riotasbc 6524 scottexs 7767 scott0s 7768 hta 7777 issubc 13990 dmdprd 15514 setinds 25348 bnj1454 28919 bnj110 28935 
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