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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 3196 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 3172 below). For example, if is a proper
?Error:
0 e. y
^
This math symbol is not active (i.e. was not declared in this scope).
class, Quine's substitution of for in If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 3172, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3171 in the form of sbc8g 3177. 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 3171 and assert that is always false when is a proper class. The theorem sbc2or 3178 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3172. The related definition dfcsb 3271 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 3170  . 2 
5  1, 2  cab 2429  . . 3 
6  3, 5  wcel 1728  . 2 
7  4, 6  wb 178  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 3172 dfsbcq2 3173 sbcex 3179 nfsbc1d 3187 nfsbcd 3190 cbvsbc 3198 sbcbid 3226 intab 4109 brab1 4288 iotacl 5487 riotasbc 6615 scottexs 7862 scott0s 7863 hta 7872 issubc 14086 dmdprd 15610 setinds 25509 sbcbi2 26849 bnj1454 29454 bnj110 29470 
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