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Definition df-sbc 3772
Description: Define the proper substitution of a class for a set.

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3781). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3801 for our definition, whose right-hand side 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 3773 below). For example, if 𝐴 is a proper class, Quine's substitution of 𝐴 for 𝑦 in 0 ∈ 𝑦 evaluates to 0 ∈ 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 𝜑, 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 3773, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3772 in the form of sbc8g 3779. 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 3772 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3780 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3773.

The related definition df-csb 3883 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]𝜑𝐴 ∈ {𝑥𝜑})

Detailed syntax breakdown of Definition df-sbc
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 cA . . 3 class 𝐴
41, 2, 3wsbc 3771 . 2 wff [𝐴 / 𝑥]𝜑
51, 2cab 2799 . . 3 class {𝑥𝜑}
63, 5wcel 2105 . 2 wff 𝐴 ∈ {𝑥𝜑}
74, 6wb 207 1 wff ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
Colors of variables: wff setvar class
This definition is referenced by:  dfsbcq  3773  dfsbcq2  3774  sbceqbid  3778  sbcex  3781  nfsbc1d  3789  nfsbcdw  3792  nfsbcd  3795  cbvsbcw  3803  cbvsbc  3805  sbcbid  3825  sbcbi2OLD  3831  intab  4899  brab1  5106  iotacl  6335  riotasbc  7121  scottexs  9305  scott0s  9306  hta  9315  issubc  17095  dmdprd  19051  sbceqbidf  30178  bnj1454  32014  bnj110  32030  setinds  32921  bj-csbsnlem  34118  rdgssun  34542  frege54cor1c  40141  frege55lem1c  40142  frege55c  40144
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