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Definition df-sbc 2910
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 2934 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 2911 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 2911, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2910 in the form of sbc8g 2916. 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 2910 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 2909 . 2  wff  [. A  /  x ]. ph
51, 2cab 2125 . . 3  class  { x  |  ph }
63, 5wcel 1480 . 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  2911  dfsbcq2  2912  sbcex  2917  nfsbc1d  2925  nfsbcd  2928  cbvsbcw  2936  cbvsbc  2937  sbcbi2  2959  sbcbid  2966  intab  3800  brab1  3975  iotacl  5111  riotasbc  5745  bdsbcALT  13087
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