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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 3179 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 3155 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 3155, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3154 in the form of sbc8g 3160. 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 3154 and assert that is always false when is a proper class. The theorem sbc2or 3161 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3155. The related definition dfcsb 3244 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 3153  . 2 
5  1, 2  cab 2421  . . 3 
6  3, 5  wcel 1725  . 2 
7  4, 6  wb 177  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 3155 dfsbcq2 3156 sbcex 3162 nfsbc1d 3170 nfsbcd 3173 cbvsbc 3181 sbcbid 3206 intab 4072 brab1 4249 iotacl 5433 riotasbc 6557 scottexs 7803 scott0s 7804 hta 7813 issubc 14027 dmdprd 15551 setinds 25397 bnj1454 29150 bnj110 29166 
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