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Definition df-sbc 2967
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 2992 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 2968 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 2968, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2967 in the form of sbc8g 2973. 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 2967 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 2974 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2968.

The related definition df-csb 3057 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 2966 . 2  wff  [. A  /  x ]. ph
51, 2cab 2244 . . 3  class  { x  |  ph }
63, 5wcel 1621 . 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  2968  dfsbcq2  2969  sbcex  2975  nfsbc1d  2983  nfsbcd  2986  cbvsbc  2994  sbcbid  3019  intab  3866  brab1  4042  iotacl  6248  riotasbc  6288  scottexs  7525  scott0s  7526  hta  7535  issubc  13674  dmdprd  15198  setinds  23503  bnj1454  27923  bnj110  27939  bnj984  28033
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