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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 3354 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 3329 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 3329, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3328 in the form of sbc8g 3335. 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 3328 and assert that is always false when is a proper class. The theorem sbc2or 3336 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3329. The related definition dfcsb 3435 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 3327  . 2 
5  1, 2  cab 2442  . . 3 
6  3, 5  wcel 1818  . 2 
7  4, 6  wb 184  1 
Colors of variables: wff setvar class 
This definition is referenced by: dfsbcq 3329 dfsbcq2 3330 sbceqbid 3334 sbcex 3337 nfsbc1d 3345 nfsbcd 3348 cbvsbc 3356 sbcbi2 3378 sbcbid 3385 intab 4317 brab1 4497 iotacl 5579 riotasbc 6273 scottexs 8326 scott0s 8327 hta 8336 issubc 15204 dmdprd 17029 sbceqbidf 27380 setinds 29210 bnj1454 33900 bnj110 33916 bjcsbsnlem 34470 frege54cor1c 37942 frege55lem1c 37943 frege55c 37945 
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