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Mirrors > Home > MPE Home > Th. List > df-sbc | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-sbc | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cA | . . 3 class 𝐴 | |
4 | 1, 2, 3 | wsbc 3468 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2637 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 2030 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 196 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
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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