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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 3691). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3708 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 3683 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 3683, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3682 in the form of sbc8g 3689. 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 3682 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3690 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3683. The related definition df-csb 3787 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 3681 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2758 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 2050 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 198 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3683 dfsbcq2 3684 sbceqbid 3688 sbcex 3691 nfsbc1d 3699 nfsbcd 3702 cbvsbc 3710 sbcbid 3730 sbcbi2OLD 3735 intab 4779 brab1 4977 iotacl 6175 riotasbc 6952 scottexs 9110 scott0s 9111 hta 9120 issubc 16963 dmdprd 18870 sbceqbidf 30032 bnj1454 31767 bnj110 31783 setinds 32549 bj-csbsnlem 33718 rdgssun 34107 frege54cor1c 39630 frege55lem1c 39631 frege55c 39633 |
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