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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 3131 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 3107 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 3107, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3106 in the form of sbc8g 3112. 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 3106 and assert that is always false when is a proper class. The theorem sbc2or 3113 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3107. The related definition dfcsb 3196 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 3105  . 2 
5  1, 2  cab 2374  . . 3 
6  3, 5  wcel 1717  . 2 
7  4, 6  wb 177  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 3107 dfsbcq2 3108 sbcex 3114 nfsbc1d 3122 nfsbcd 3125 cbvsbc 3133 sbcbid 3158 intab 4023 brab1 4199 iotacl 5382 riotasbc 6502 scottexs 7745 scott0s 7746 hta 7755 issubc 13963 dmdprd 15487 setinds 25159 bnj1454 28552 bnj110 28568 
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