Metamath Proof Explorer 
< Previous
Next >
Nearby theorems 

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 3155 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 3131 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 3131, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 3130 in the form of sbc8g 3136. 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 3130 and assert that is always false when is a proper class. The theorem sbc2or 3137 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3131. The related definition dfcsb 3220 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 3129  . 2 
5  1, 2  cab 2398  . . 3 
6  3, 5  wcel 1721  . 2 
7  4, 6  wb 177  1 
Colors of variables: wff set class 
This definition is referenced by: dfsbcq 3131 dfsbcq2 3132 sbcex 3138 nfsbc1d 3146 nfsbcd 3149 cbvsbc 3157 sbcbid 3182 intab 4048 brab1 4225 iotacl 5408 riotasbc 6532 scottexs 7775 scott0s 7776 hta 7785 issubc 13998 dmdprd 15522 setinds 25356 bnj1454 28931 bnj110 28947 
Copyright terms: Public domain  W3C validator 