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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 3144 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 3120 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 3120, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3119 in the form of sbc8g 3125. 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 3119 and assert that is always false when is a proper class. The theorem sbc2or 3126 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3120. The related definition dfcsb 3209 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 3118  . 2 
5  1, 2  cab 2387  . . 3 
6  3, 5  wcel 1721  . 2 
7  4, 6  wb 177  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 3120 dfsbcq2 3121 sbcex 3127 nfsbc1d 3135 nfsbcd 3138 cbvsbc 3146 sbcbid 3171 intab 4036 brab1 4212 iotacl 5395 riotasbc 6515 scottexs 7758 scott0s 7759 hta 7768 issubc 13976 dmdprd 15500 setinds 25317 bnj1454 28859 bnj110 28875 
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