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Definition df-sbc 3154
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 3179 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 3155 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 3155, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3154 in the form of sbc8g 3160. 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 3154 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3161 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3155.

The related definition df-csb 3244 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  set  x
3 cA . . 3  class  A
41, 2, 3wsbc 3153 . 2  wff  [. A  /  x ]. ph
51, 2cab 2421 . . 3  class  { x  |  ph }
63, 5wcel 1725 . 2  wff  A  e. 
{ x  |  ph }
74, 6wb 177 1  wff  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  3155  dfsbcq2  3156  sbcex  3162  nfsbc1d  3170  nfsbcd  3173  cbvsbc  3181  sbcbid  3206  intab  4072  brab1  4249  iotacl  5433  riotasbc  6557  scottexs  7803  scott0s  7804  hta  7813  issubc  14027  dmdprd  15551  setinds  25397  bnj1454  29150  bnj110  29166
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