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Definition df-sbc 3327
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 3353 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 3328 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 3328, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3327 in the form of sbc8g 3334. 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 3327 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3335 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3328.

The related definition df-csb 3431 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 3326 . 2  wff  [. A  /  x ]. ph
51, 2cab 2447 . . 3  class  { x  |  ph }
63, 5wcel 1762 . 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  3328  dfsbcq2  3329  sbceqbid  3333  sbcex  3336  nfsbc1d  3344  nfsbcd  3347  cbvsbc  3355  sbcbi2  3377  sbcbid  3384  intab  4307  brab1  4487  iotacl  5567  riotasbc  6254  scottexs  8296  scott0s  8297  hta  8306  issubc  15056  dmdprd  16815  sbceqbidf  27044  setinds  28775  bnj1454  32856  bnj110  32872  bj-csbsnlem  33428  frege54cor1c  36784  frege55lem1c  36785  frege55c  36787  frege58newc  36790
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