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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 (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.) |
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 3747 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2800 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 2114 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 209 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
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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