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Mirrors > Home > MPE Home > Th. List > dfsbc  Structured version 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 3353 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 3328 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 3328, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3327 in the form of sbc8g 3334. 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 3327 and assert that is always false when is a proper class. The theorem sbc2or 3335 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3328. The related definition dfcsb 3431 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 3326  . 2 
5  1, 2  cab 2447  . . 3 
6  3, 5  wcel 1762  . 2 
7  4, 6  wb 184  1 
Colors of variables: wff setvar class 
This definition is referenced by: dfsbcq 3328 dfsbcq2 3329 sbceqbid 3333 sbcex 3336 nfsbc1d 3344 nfsbcd 3347 cbvsbc 3355 sbcbi2 3377 sbcbid 3384 intab 4307 brab1 4487 iotacl 5567 riotasbc 6254 scottexs 8296 scott0s 8297 hta 8306 issubc 15056 dmdprd 16815 sbceqbidf 27044 setinds 28775 bnj1454 32856 bnj110 32872 bjcsbsnlem 33428 frege54cor1c 36784 frege55lem1c 36785 frege55c 36787 frege58newc 36790 
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