Definition df-sbc 3744

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3753). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3774 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 3745 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 3745, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3744 in the form of sbc8g 3751. 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 3744 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3752 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3745.

The related definition df-csb 3853 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 3743	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2739	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2141	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 209	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3745  dfsbcq2  3746  sbceqbid  3750  sbcex  3753  nfsbc1d  3761  nfsbcdw  3764  nfsbcd  3767  sbc5  3771  sbc6g  3773  cbvsbcw  3776  cbvsbcvw  3777  cbvsbc  3778  sbcieg  3782  sbcied  3786  sbcbid  3797  sbcbi2  3801  sbcimdv  3811  sbcg  3815  intab  4942  brab1  5158  iotacl  6522  riotasbc  7385  setinds  9717  scottexs  9860  scott0s  9861  hta  9882  issubc  17891  dmdprd  20069  sbceqbidf  32799  bnj1454  35196  bnj110  35212  sbceqbii  36647  cbvsbcvw2  36686  cbvsbcdavw  36713  cbvsbcdavw2  36714  bj-csbsnlem  37482  rdgssun  37968  frege54cor1c  44589  frege55lem1c  44590  frege55c  44592