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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 3019 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 2995 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 2995, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 2994 in the form of sbc8g 3000. 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 2994 and assert that is always false when is a proper class. The theorem sbc2or 3001 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2995. The related definition dfcsb 3084 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 2993  . 2 
5  1, 2  cab 2271  . . 3 
6  3, 5  wcel 1685  . 2 
7  4, 6  wb 178  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 2995 dfsbcq2 2996 sbcex 3002 nfsbc1d 3010 nfsbcd 3013 cbvsbc 3021 sbcbid 3046 intab 3894 brab1 4070 iotacl 6276 riotasbc 6316 scottexs 7553 scott0s 7554 hta 7563 issubc 13707 dmdprd 15231 setinds 23536 bnj1454 28142 bnj110 28158 bnj984 28252 
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