Definition df-sbc 3805

Description: Define the proper substitution of a class for a set.

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3814). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3836 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 3806 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 3806, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3805 in the form of sbc8g 3812. 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 3805 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3813 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3806.

The related definition df-csb 3922 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.)

Assertion

Ref	Expression
df-sbc	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

Detailed syntax breakdown of Definition df-sbc

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3		cA	. . 3 class 𝐴
4	1, 2, 3	wsbc 3804	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2717	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2108	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 206	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3806  dfsbcq2  3807  sbceqbid  3811  sbcex  3814  nfsbc1d  3822  nfsbcdw  3825  nfsbcd  3828  sbc5  3832  sbc6g  3834  cbvsbcw  3838  cbvsbcvw  3839  cbvsbc  3840  sbcieg  3845  sbcied  3850  sbcbid  3863  sbcbi2  3867  sbcimdv  3878  sbcg  3883  intab  5002  brab1  5214  iotacl  6559  riotasbc  7423  scottexs  9956  scott0s  9957  hta  9966  issubc  17899  dmdprd  20042  sbceqbidf  32515  bnj1454  34818  bnj110  34834  setinds  35742  sbceqbii  36155  cbvsbcvw2  36196  cbvsbcdavw  36223  cbvsbcdavw2  36224  bj-csbsnlem  36869  rdgssun  37344  frege54cor1c  43877  frege55lem1c  43878  frege55c  43880