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Definition df-sbc 3634
Description: Define the proper substitution of a class for a set.

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3643). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3660 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 3635 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 3635, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3634 in the form of sbc8g 3641. 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 3634 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3642 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3635.

The related definition df-csb 3729 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 3633 . 2 wff [𝐴 / 𝑥]𝜑
51, 2cab 2792 . . 3 class {𝑥𝜑}
63, 5wcel 2156 . 2 wff 𝐴 ∈ {𝑥𝜑}
74, 6wb 197 1 wff ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
Colors of variables: wff setvar class
This definition is referenced by:  dfsbcq  3635  dfsbcq2  3636  sbceqbid  3640  sbcex  3643  nfsbc1d  3651  nfsbcd  3654  cbvsbc  3662  sbcbi2  3682  sbcbid  3687  intab  4699  brab1  4892  iotacl  6083  riotasbc  6846  scottexs  8993  scott0s  8994  hta  9003  issubc  16695  dmdprd  18595  sbceqbidf  29645  bnj1454  31230  bnj110  31246  setinds  31998  bj-csbsnlem  33201  frege54cor1c  38703  frege55lem1c  38704  frege55c  38706
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