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Definition df-sbc 3306
Description: Define the proper substitution of a class for a set.

When  A is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3332 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 3307 below). For example, if  A is a proper class, Quine's substitution of 
A for  y in  0  e.  y evaluates to  0  e.  A rather than our falsehood. (This can be seen by substituting  A,  y, and  0 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  ph, 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 3307, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3306 in the form of sbc8g 3313. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of  A in every use of this definition) we allow direct reference to df-sbc 3306 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3314 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3307.

The related definition df-csb 3402 defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.)

Assertion
Ref Expression
df-sbc  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)

Detailed syntax breakdown of Definition df-sbc
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  setvar  x
3 cA . . 3  class  A
41, 2, 3wsbc 3305 . 2  wff  [. A  /  x ]. ph
51, 2cab 2414 . . 3  class  { x  |  ph }
63, 5wcel 1870 . 2  wff  A  e. 
{ x  |  ph }
74, 6wb 187 1  wff  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
Colors of variables: wff setvar class
This definition is referenced by:  dfsbcq  3307  dfsbcq2  3308  sbceqbid  3312  sbcex  3315  nfsbc1d  3323  nfsbcd  3326  cbvsbc  3334  sbcbi2  3354  sbcbid  3359  intab  4289  brab1  4471  iotacl  5588  riotasbc  6282  scottexs  8357  scott0s  8358  hta  8367  issubc  15691  dmdprd  17565  sbceqbidf  27952  bnj1454  29441  bnj110  29457  setinds  30211  bj-csbsnlem  31255  frege54cor1c  36148  frege55lem1c  36149  frege55c  36151
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