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Definition df-sbc 3130
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 3155 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 3131 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 3131, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3130 in the form of sbc8g 3136. 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 3130 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3137 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3131.

The related definition df-csb 3220 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 3129 . 2  wff  [. A  /  x ]. ph
51, 2cab 2398 . . 3  class  { x  |  ph }
63, 5wcel 1721 . 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  3131  dfsbcq2  3132  sbcex  3138  nfsbc1d  3146  nfsbcd  3149  cbvsbc  3157  sbcbid  3182  intab  4048  brab1  4225  iotacl  5408  riotasbc  6532  scottexs  7775  scott0s  7776  hta  7785  issubc  13998  dmdprd  15522  setinds  25356  bnj1454  28931  bnj110  28947
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