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Definition df-sbc 2936
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 2961 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 2937 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 2937, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2936 in the form of sbc8g 2942. 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 2936 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 2943 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2937.

The related definition df-csb 3024 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 2935 . 2  wff  [. A  /  x ]. ph
51, 2cab 2242 . . 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  2937  dfsbcq2  2938  sbcex  2944  nfsbc1d  2952  nfsbcd  2955  cbvsbc  2963  sbcbid  2988  intab  3833  brab1  4008  iotacl  6213  riotasbc  6253  scottexs  7490  scott0s  7491  hta  7500  issubc  13639  dmdprd  15163  setinds  23468  bnj1454  27886  bnj110  27902  bnj984  27996
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