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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 3782). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3804 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 3774 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 3774, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3773 in the form of sbc8g 3780. 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 3773 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. Theorem sbc2or 3781 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3774. The related definition df-csb 3889 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 3772 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2703 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 2098 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 205 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3774 dfsbcq2 3775 sbceqbid 3779 sbcex 3782 nfsbc1d 3790 nfsbcdw 3793 nfsbcd 3796 sbc5 3800 sbc6g 3802 cbvsbcw 3806 cbvsbcvw 3807 cbvsbc 3808 sbcieg 3812 sbcied 3817 sbcbid 3830 sbcbi2 3834 sbcimdv 3846 sbcg 3851 intab 4975 brab1 5189 iotacl 6522 riotasbc 7379 scottexs 9881 scott0s 9882 hta 9891 issubc 17792 dmdprd 19918 sbceqbidf 32232 bnj1454 34382 bnj110 34398 setinds 35283 bj-csbsnlem 36290 rdgssun 36766 frege54cor1c 43223 frege55lem1c 43224 frege55c 43226 |
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