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Definition df-sbc 3748
Description: Define the proper substitution of a class for a set.

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3757). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3777 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 3749 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 3749, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3748 in the form of sbc8g 3755. 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 3748 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3756 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3749.

The related definition df-csb 3856 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 3747 . 2 wff [𝐴 / 𝑥]𝜑
51, 2cab 2800 . . 3 class {𝑥𝜑}
63, 5wcel 2114 . 2 wff 𝐴 ∈ {𝑥𝜑}
74, 6wb 209 1 wff ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
Colors of variables: wff setvar class
This definition is referenced by:  dfsbcq  3749  dfsbcq2  3750  sbceqbid  3754  sbcex  3757  nfsbc1d  3765  nfsbcdw  3768  nfsbcd  3771  cbvsbcw  3779  cbvsbc  3781  sbcbid  3800  sbcbi2  3805  intab  4881  brab1  5090  iotacl  6320  riotasbc  7116  scottexs  9304  scott0s  9305  hta  9314  issubc  17096  dmdprd  19111  sbceqbidf  30255  bnj1454  32188  bnj110  32204  setinds  33097  bj-csbsnlem  34305  rdgssun  34756  frege54cor1c  40547  frege55lem1c  40548  frege55c  40550
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