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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 3017 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 2993 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 2993, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 2992 in the form of sbc8g 2998. 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 2992 and assert that is always false when is a proper class. The theorem sbc2or 2999 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2993. The related definition dfcsb 3082 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 2991  . 2 
5  1, 2  cab 2269  . . 3 
6  3, 5  wcel 1684  . 2 
7  4, 6  wb 176  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 2993 dfsbcq2 2994 sbcex 3000 nfsbc1d 3008 nfsbcd 3011 cbvsbc 3019 sbcbid 3044 intab 3892 brab1 4068 iotacl 5242 riotasbc 6320 scottexs 7557 scott0s 7558 hta 7567 issubc 13712 dmdprd 15236 setinds 24134 bnj1454 28874 bnj110 28890 bnj984 28984 
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