Definition df-sbc 3737

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3746). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3767 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 3738 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 3738, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3737 in the form of sbc8g 3744. 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 3737 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3745 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3738.

The related definition df-csb 3846 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 3736	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2709	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2111	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 206	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3738  dfsbcq2  3739  sbceqbid  3743  sbcex  3746  nfsbc1d  3754  nfsbcdw  3757  nfsbcd  3760  sbc5  3764  sbc6g  3766  cbvsbcw  3769  cbvsbcvw  3770  cbvsbc  3771  sbcieg  3776  sbcied  3780  sbcbid  3791  sbcbi2  3795  sbcimdv  3805  sbcg  3809  intab  4926  brab1  5137  iotacl  6467  riotasbc  7321  setinds  9639  scottexs  9780  scott0s  9781  hta  9790  issubc  17742  dmdprd  19912  sbceqbidf  32466  bnj1454  34854  bnj110  34870  sbceqbii  36233  cbvsbcvw2  36272  cbvsbcdavw  36299  cbvsbcdavw2  36300  bj-csbsnlem  36945  rdgssun  37420  frege54cor1c  43956  frege55lem1c  43957  frege55c  43959