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Definition df-sbc 2992
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 3017 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 2993 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 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 2993, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2992 in the form of sbc8g 2998. 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 2992 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 2999 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2993.

The related definition df-csb 3082 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 2991 . 2  wff  [. A  /  x ]. ph
51, 2cab 2269 . . 3  class  { x  |  ph }
63, 5wcel 1684 . 2  wff  A  e. 
{ x  |  ph }
74, 6wb 176 1  wff  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  2993  dfsbcq2  2994  sbcex  3000  nfsbc1d  3008  nfsbcd  3011  cbvsbc  3019  sbcbid  3044  intab  3892  brab1  4068  iotacl  5242  riotasbc  6320  scottexs  7557  scott0s  7558  hta  7567  issubc  13712  dmdprd  15236  setinds  24134  bnj1454  28874  bnj110  28890  bnj984  28984
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