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Definition df-sbc 3469
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 3495 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 3470 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 3470, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3469 in the form of sbc8g 3476. 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 3469 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3477 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3470.

The related definition df-csb 3567 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 3468 . 2 wff [𝐴 / 𝑥]𝜑
51, 2cab 2637 . . 3 class {𝑥𝜑}
63, 5wcel 2030 . 2 wff 𝐴 ∈ {𝑥𝜑}
74, 6wb 196 1 wff ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
Colors of variables: wff setvar class
This definition is referenced by:  dfsbcq  3470  dfsbcq2  3471  sbceqbid  3475  sbcex  3478  nfsbc1d  3486  nfsbcd  3489  cbvsbc  3497  sbcbi2  3517  sbcbid  3522  intab  4539  brab1  4733  iotacl  5912  riotasbc  6666  scottexs  8788  scott0s  8789  hta  8798  issubc  16542  dmdprd  18443  sbceqbidf  29449  bnj1454  31038  bnj110  31054  setinds  31807  bj-csbsnlem  33023  frege54cor1c  38526  frege55lem1c  38527  frege55c  38529
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