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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 3324 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 3299 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 3299, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3298 in the form of sbc8g 3305. 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 3298 and assert that is always false when is a proper class. The theorem sbc2or 3306 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3299. The related definition dfcsb 3402 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 3297  . 2 
5  1, 2  cab 2439  . . 3 
6  3, 5  wcel 1758  . 2 
7  4, 6  wb 184  1 
Colors of variables: wff setvar class 
This definition is referenced by: dfsbcq 3299 dfsbcq2 3300 sbceqbid 3304 sbcex 3307 nfsbc1d 3315 nfsbcd 3318 cbvsbc 3326 sbcbi2 3348 sbcbid 3355 intab 4275 brab1 4454 iotacl 5523 riotasbc 6199 scottexs 8231 scott0s 8232 hta 8241 issubc 14907 dmdprd 16655 sbceqbidf 26334 setinds 28047 bnj1454 32678 bnj110 32694 bjcsbsnlem 33250 frege54cor1c 36596 frege55lem1c 36597 frege55c 36599 frege58newc 36602 
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