Definition df-sbc 3775

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3784). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3806 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 3776 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 3776, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3775 in the form of sbc8g 3782. 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 3775 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3783 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3776.

The related definition df-csb 3890 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 3774	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2704	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2099	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 205	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3776  dfsbcq2  3777  sbceqbid  3781  sbcex  3784  nfsbc1d  3792  nfsbcdw  3795  nfsbcd  3798  sbc5  3802  sbc6g  3804  cbvsbcw  3808  cbvsbcvw  3809  cbvsbc  3810  sbcieg  3814  sbcied  3819  sbcbid  3832  sbcbi2  3836  sbcimdv  3847  sbcg  3852  intab  4976  brab1  5190  iotacl  6528  riotasbc  7389  scottexs  9896  scott0s  9897  hta  9906  issubc  17806  dmdprd  19939  sbceqbidf  32258  bnj1454  34396  bnj110  34412  setinds  35297  bj-csbsnlem  36304  rdgssun  36780  frege54cor1c  43258  frege55lem1c  43259  frege55c  43261