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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 2961 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 2937 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 2937, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 2936 in the form of sbc8g 2942. 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 2936 and assert that is always false when is a proper class. The theorem sbc2or 2943 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2937. The related definition dfcsb 3024 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 2935  . 2 
5  1, 2  cab 2242  . . 3 
6  3, 5  wcel 1621  . 2 
7  4, 6  wb 178  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 2937 dfsbcq2 2938 sbcex 2944 nfsbc1d 2952 nfsbcd 2955 cbvsbc 2963 sbcbid 2988 intab 3833 brab1 4008 iotacl 6213 riotasbc 6253 scottexs 7490 scott0s 7491 hta 7500 issubc 13639 dmdprd 15163 setinds 23468 bnj1454 27886 bnj110 27902 bnj984 27996 
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