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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 2992 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 2968 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 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 2968, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 2967 in the form of sbc8g 2973. 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 2967 and assert that is always false when is a proper class. The theorem sbc2or 2974 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2968. The related definition dfcsb 3057 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 2966  . 2 
5  1, 2  cab 2244  . . 3 
6  3, 5  wcel 1621  . 2 
7  4, 6  wb 178  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 2968 dfsbcq2 2969 sbcex 2975 nfsbc1d 2983 nfsbcd 2986 cbvsbc 2994 sbcbid 3019 intab 3866 brab1 4042 iotacl 6248 riotasbc 6288 scottexs 7525 scott0s 7526 hta 7535 issubc 13674 dmdprd 15198 setinds 23503 bnj1454 27923 bnj110 27939 bnj984 28033 
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