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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 3643). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3660 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 3635 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 3635, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3634 in the form of sbc8g 3641. 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 3634 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3642 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3635. The related definition df-csb 3729 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 3633 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2792 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 2156 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 197 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3635 dfsbcq2 3636 sbceqbid 3640 sbcex 3643 nfsbc1d 3651 nfsbcd 3654 cbvsbc 3662 sbcbi2 3682 sbcbid 3687 intab 4699 brab1 4892 iotacl 6083 riotasbc 6846 scottexs 8993 scott0s 8994 hta 9003 issubc 16695 dmdprd 18595 sbceqbidf 29645 bnj1454 31230 bnj110 31246 setinds 31998 bj-csbsnlem 33201 frege54cor1c 38703 frege55lem1c 38704 frege55c 38706 |
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