Definition df-sbc 3747

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3756). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3777 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 3748 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 3748, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3747 in the form of sbc8g 3754. 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 3747 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3755 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3748.

The related definition df-csb 3855 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 3746	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2742	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2144	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 208	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3748  dfsbcq2  3749  sbceqbid  3753  sbcex  3756  nfsbc1d  3764  nfsbcdw  3767  nfsbcd  3770  sbc5  3774  sbc6g  3776  cbvsbcw  3779  cbvsbcvw  3780  cbvsbc  3781  sbcieg  3785  sbcied  3789  sbcbid  3800  sbcbi2  3804  sbcimdv  3814  sbcg  3818  intab  4938  brab1  5150  iotacl  6509  riotasbc  7373  setinds  9706  scottexs  9847  scott0s  9848  hta  9857  issubc  17870  dmdprd  20042  sbceqbidf  32688  bnj1454  35139  bnj110  35155  sbceqbii  36556  cbvsbcvw2  36595  cbvsbcdavw  36622  cbvsbcdavw2  36623  bj-csbsnlem  37393  rdgssun  37877  frege54cor1c  44496  frege55lem1c  44497  frege55c  44499