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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 3030 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 3006 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 3006, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3005 in the form of sbc8g 3011. 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 3005 and assert that is always false when is a proper class. The theorem sbc2or 3012 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3006. The related definition dfcsb 3095 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 3004  . 2 
5  1, 2  cab 2282  . . 3 
6  3, 5  wcel 1696  . 2 
7  4, 6  wb 176  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 3006 dfsbcq2 3007 sbcex 3013 nfsbc1d 3021 nfsbcd 3024 cbvsbc 3032 sbcbid 3057 intab 3908 brab1 4084 iotacl 5258 riotasbc 6336 scottexs 7573 scott0s 7574 hta 7583 issubc 13728 dmdprd 15252 setinds 24205 bnj1454 29190 bnj110 29206 bnj984 29300 
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