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Definition df-sbc 3106
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 3131 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 3107 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 3107, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3106 in the form of sbc8g 3112. 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 3106 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3113 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3107.

The related definition df-csb 3196 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 3105 . 2  wff  [. A  /  x ]. ph
51, 2cab 2374 . . 3  class  { x  |  ph }
63, 5wcel 1717 . 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  3107  dfsbcq2  3108  sbcex  3114  nfsbc1d  3122  nfsbcd  3125  cbvsbc  3133  sbcbid  3158  intab  4023  brab1  4199  iotacl  5382  riotasbc  6502  scottexs  7745  scott0s  7746  hta  7755  issubc  13963  dmdprd  15487  setinds  25159  bnj1454  28552  bnj110  28568
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