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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 3781). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3801 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 3773 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 3773, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3772 in the form of sbc8g 3779. 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 3772 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3780 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3773. The related definition df-csb 3883 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 3771 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2799 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 2105 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 207 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3773 dfsbcq2 3774 sbceqbid 3778 sbcex 3781 nfsbc1d 3789 nfsbcdw 3792 nfsbcd 3795 cbvsbcw 3803 cbvsbc 3805 sbcbid 3825 sbcbi2OLD 3831 intab 4899 brab1 5106 iotacl 6335 riotasbc 7121 scottexs 9305 scott0s 9306 hta 9315 issubc 17095 dmdprd 19051 sbceqbidf 30178 bnj1454 32014 bnj110 32030 setinds 32921 bj-csbsnlem 34118 rdgssun 34542 frege54cor1c 40141 frege55lem1c 40142 frege55c 40144 |
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