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Definition df-sbc 3163
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 3188 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 3164 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 3164, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3163 in the form of sbc8g 3169. 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 3163 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3170 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3164.

The related definition df-csb 3253 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 3162 . 2  wff  [. A  /  x ]. ph
51, 2cab 2423 . . 3  class  { x  |  ph }
63, 5wcel 1726 . 2  wff  A  e. 
{ x  |  ph }
74, 6wb 178 1  wff  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  3164  dfsbcq2  3165  sbcex  3171  nfsbc1d  3179  nfsbcd  3182  cbvsbc  3190  sbcbid  3215  intab  4081  brab1  4258  iotacl  5442  riotasbc  6566  scottexs  7812  scott0s  7813  hta  7822  issubc  14036  dmdprd  15560  setinds  25406  bnj1454  29214  bnj110  29230
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