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 2947 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 2923 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 2923, which holds for both our definition and Quine's, and from which we can derive a weaker version of dfsbc 2922 in the form of sbc8g 2928. 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 2922 and assert that is always false when is a proper class. The theorem sbc2or 2929 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2923. The related definition dfcsb 3010 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 2921  . 2 
5  1, 2  cab 2239  . . 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 2923 dfsbcq2 2924 sbcex 2930 nfsbc1d 2938 nfsbcd 2941 cbvsbc 2949 sbcbid 2974 intab 3790 brab1 3965 iotacl 6166 riotasbc 6206 scottexs 7441 scott0s 7442 hta 7451 issubc 13556 dmdprd 15071 setinds 23302 bnj1454 27563 bnj110 27579 bnj984 27673 
Copyright terms: Public domain  W3C validator 