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Definition df-sbc 3314
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 3340 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 3315 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 3315, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3314 in the form of sbc8g 3321. 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 3314 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3322 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3315.

The related definition df-csb 3421 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 3313 . 2  wff  [. A  /  x ]. ph
51, 2cab 2428 . . 3  class  { x  |  ph }
63, 5wcel 1804 . 2  wff  A  e. 
{ x  |  ph }
74, 6wb 184 1  wff  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
Colors of variables: wff setvar class
This definition is referenced by:  dfsbcq  3315  dfsbcq2  3316  sbceqbid  3320  sbcex  3323  nfsbc1d  3331  nfsbcd  3334  cbvsbc  3342  sbcbi2  3364  sbcbid  3371  intab  4302  brab1  4482  iotacl  5564  riotasbc  6258  scottexs  8308  scott0s  8309  hta  8318  issubc  15186  dmdprd  17008  sbceqbidf  27358  setinds  29186  bnj1454  33768  bnj110  33784  bj-csbsnlem  34353  frege54cor1c  37746  frege55lem1c  37747  frege55c  37749
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