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Definition df-sbc 3253
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 3279 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 3254 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 3254, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3253 in the form of sbc8g 3260. 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 3253 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3261 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3254.

The related definition df-csb 3349 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 3252 . 2  wff  [. A  /  x ]. ph
51, 2cab 2367 . . 3  class  { x  |  ph }
63, 5wcel 1826 . 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  3254  dfsbcq2  3255  sbceqbid  3259  sbcex  3262  nfsbc1d  3270  nfsbcd  3273  cbvsbc  3281  sbcbi2  3301  sbcbid  3306  intab  4230  brab1  4412  iotacl  5483  riotasbc  6173  scottexs  8218  scott0s  8219  hta  8228  issubc  15241  dmdprd  17142  sbceqbidf  27497  setinds  29375  bnj1454  34247  bnj110  34263  bj-csbsnlem  34817  frege54cor1c  38414  frege55lem1c  38415  frege55c  38417
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