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Definition df-sbc 3402
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 3428 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 3403 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 3403, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3402 in the form of sbc8g 3409. 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 3402 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3410 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3403.

The related definition df-csb 3499 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 3401 . 2 wff [𝐴 / 𝑥]𝜑
51, 2cab 2595 . . 3 class {𝑥𝜑}
63, 5wcel 1976 . 2 wff 𝐴 ∈ {𝑥𝜑}
74, 6wb 194 1 wff ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
Colors of variables: wff setvar class
This definition is referenced by:  dfsbcq  3403  dfsbcq2  3404  sbceqbid  3408  sbcex  3411  nfsbc1d  3419  nfsbcd  3422  cbvsbc  3430  sbcbi2  3450  sbcbid  3455  intab  4436  brab1  4624  iotacl  5776  riotasbc  6503  scottexs  8610  scott0s  8611  hta  8620  issubc  16266  dmdprd  18168  sbceqbidf  28498  bnj1454  29959  bnj110  29975  setinds  30720  bj-csbsnlem  31873  frege54cor1c  37012  frege55lem1c  37013  frege55c  37015
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