Definition df-sbc 3725

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3734). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3755 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 3726 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 3726, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3725 in the form of sbc8g 3732. 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 3725 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3733 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3726.

The related definition df-csb 3833 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 3724	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2719	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2121	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 208	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3726  dfsbcq2  3727  sbceqbid  3731  sbcex  3734  nfsbc1d  3742  nfsbcdw  3745  nfsbcd  3748  sbc5  3752  sbc6g  3754  cbvsbcw  3757  cbvsbcvw  3758  cbvsbc  3759  sbcieg  3763  sbcied  3767  sbcbid  3778  sbcbi2  3782  sbcimdv  3792  sbcg  3796  intab  4910  brab1  5122  iotacl  6474  riotasbc  7334  setinds  9665  scottexs  9806  scott0s  9807  hta  9816  issubc  17797  dmdprd  19969  sbceqbidf  32576  bnj1454  35037  bnj110  35053  sbceqbii  36432  cbvsbcvw2  36471  cbvsbcdavw  36498  cbvsbcdavw2  36499  bj-csbsnlem  37269  rdgssun  37753  frege54cor1c  44372  frege55lem1c  44373  frege55c  44375