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Definition df-sbc 3005
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 3030 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 3006 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 3006, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3005 in the form of sbc8g 3011. 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 3005 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3012 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3006.

The related definition df-csb 3095 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 3004 . 2  wff  [. A  /  x ]. ph
51, 2cab 2282 . . 3  class  { x  |  ph }
63, 5wcel 1696 . 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  3006  dfsbcq2  3007  sbcex  3013  nfsbc1d  3021  nfsbcd  3024  cbvsbc  3032  sbcbid  3057  intab  3908  brab1  4084  iotacl  5258  riotasbc  6336  scottexs  7573  scott0s  7574  hta  7583  issubc  13728  dmdprd  15252  setinds  24205  bnj1454  29190  bnj110  29206  bnj984  29300
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