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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 3189 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 3165 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 3165, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3164 in the form of sbc8g 3170. 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 3164 and assert that is always false when is a proper class. The theorem sbc2or 3171 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3165. The related definition dfcsb 3254 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 3163  . 2 
5  1, 2  cab 2424  . . 3 
6  3, 5  wcel 1726  . 2 
7  4, 6  wb 178  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 3165 dfsbcq2 3166 sbcex 3172 nfsbc1d 3180 nfsbcd 3183 cbvsbc 3191 sbcbid 3216 intab 4082 brab1 4260 iotacl 5444 riotasbc 6568 scottexs 7816 scott0s 7817 hta 7826 issubc 14040 dmdprd 15564 setinds 25410 bnj1454 29287 bnj110 29303 
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