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Definition df-sbc 2883
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 2907 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 2884 below). 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 2884, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2883 in the form of sbc8g 2889. 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 2883 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The related definition df-csb 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  setvar  x
3 cA . . 3  class  A
41, 2, 3wsbc 2882 . 2  wff  [. A  /  x ]. ph
51, 2cab 2103 . . 3  class  { x  |  ph }
63, 5wcel 1465 . 2  wff  A  e. 
{ x  |  ph }
74, 6wb 104 1  wff  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  2884  dfsbcq2  2885  sbcex  2890  nfsbc1d  2898  nfsbcd  2901  cbvsbc  2909  sbcbi2  2931  sbcbid  2938  intab  3770  brab1  3945  iotacl  5081  riotasbc  5713  bdsbcALT  12953
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