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Definition df-sbc 3243
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 3269 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 3244 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 syntactic 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 3244, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3243 in the form of sbc8g 3250. 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 3243 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3251 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3244.

The related definition df-csb 3339 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 3242 . 2  wff  [. A  /  x ]. ph
51, 2cab 2414 . . 3  class  { x  |  ph }
63, 5wcel 1872 . 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  3244  dfsbcq2  3245  sbceqbid  3249  sbcex  3252  nfsbc1d  3260  nfsbcd  3263  cbvsbc  3271  sbcbi2  3291  sbcbid  3296  intab  4229  brab1  4412  iotacl  5531  riotasbc  6226  scottexs  8310  scott0s  8311  hta  8320  issubc  15683  dmdprd  17573  sbceqbidf  28059  bnj1454  29605  bnj110  29621  setinds  30375  bj-csbsnlem  31416  frege54cor1c  36424  frege55lem1c  36425  frege55c  36427
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