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Definition df-sbc 3280
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 3306 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 3281 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 syntactic 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 3281, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3280 in the form of sbc8g 3287. 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 3280 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3288 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3281.

The related definition df-csb 3376 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 3279 . 2  wff  [. A  /  x ]. ph
51, 2cab 2448 . . 3  class  { x  |  ph }
63, 5wcel 1898 . 2  wff  A  e. 
{ x  |  ph }
74, 6wb 189 1  wff  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
Colors of variables: wff setvar class
This definition is referenced by:  dfsbcq  3281  dfsbcq2  3282  sbceqbid  3286  sbcex  3289  nfsbc1d  3297  nfsbcd  3300  cbvsbc  3308  sbcbi2  3328  sbcbid  3333  intab  4279  brab1  4462  iotacl  5588  riotasbc  6292  scottexs  8384  scott0s  8385  hta  8394  issubc  15789  dmdprd  17679  sbceqbidf  28166  bnj1454  29702  bnj110  29718  setinds  30473  bj-csbsnlem  31550  frege54cor1c  36556  frege55lem1c  36557  frege55c  36559
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