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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 3595 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 3570 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 3570, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3569 in the form of sbc8g 3576. 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 3569 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3577 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3570. The related definition df-csb 3667 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 3568 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2738 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 2131 | . 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 3570 dfsbcq2 3571 sbceqbid 3575 sbcex 3578 nfsbc1d 3586 nfsbcd 3589 cbvsbc 3597 sbcbi2 3617 sbcbid 3622 intab 4651 brab1 4844 iotacl 6027 riotasbc 6781 scottexs 8915 scott0s 8916 hta 8925 issubc 16688 dmdprd 18589 sbceqbidf 29622 bnj1454 31211 bnj110 31227 setinds 31980 bj-csbsnlem 33196 frege54cor1c 38703 frege55lem1c 38704 frege55c 38706 |
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