![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3730). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3750 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 3722 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 3722, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3721 in the form of sbc8g 3728. 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 3721 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3729 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3722. The related definition df-csb 3829 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 3720 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2776 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 2111 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 209 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3722 dfsbcq2 3723 sbceqbid 3727 sbcex 3730 nfsbc1d 3738 nfsbcdw 3741 nfsbcd 3744 cbvsbcw 3752 cbvsbc 3754 sbcbid 3773 sbcbi2 3778 intab 4868 brab1 5078 iotacl 6310 riotasbc 7111 scottexs 9300 scott0s 9301 hta 9310 issubc 17097 dmdprd 19113 sbceqbidf 30257 bnj1454 32224 bnj110 32240 setinds 33136 bj-csbsnlem 34344 rdgssun 34795 frege54cor1c 40616 frege55lem1c 40617 frege55c 40619 |
Copyright terms: Public domain | W3C validator |