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Definition df-sbc 2922
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 2947 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 2923 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 2923, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2922 in the form of sbc8g 2928. 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 2922 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 2929 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2923.

The related definition df-csb 3010 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 2921 . 2  wff  [. A  /  x ]. ph
51, 2cab 2239 . . 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  2923  dfsbcq2  2924  sbcex  2930  nfsbc1d  2938  nfsbcd  2941  cbvsbc  2949  sbcbid  2974  intab  3790  brab1  3965  iotacl  6166  riotasbc  6206  scottexs  7441  scott0s  7442  hta  7451  issubc  13556  dmdprd  15071  setinds  23302  bnj1454  27660  bnj110  27676  bnj984  27770
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