Definition df-sbc 3726

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3735). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3756 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 3727 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 3727, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3726 in the form of sbc8g 3733. 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 3726 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3734 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3727.

The related definition df-csb 3834 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 3725	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2713	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2114	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 206	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3727  dfsbcq2  3728  sbceqbid  3732  sbcex  3735  nfsbc1d  3743  nfsbcdw  3746  nfsbcd  3749  sbc5  3753  sbc6g  3755  cbvsbcw  3758  cbvsbcvw  3759  cbvsbc  3760  sbcieg  3764  sbcied  3768  sbcbid  3779  sbcbi2  3783  sbcimdv  3793  sbcg  3797  intab  4910  brab1  5122  iotacl  6473  riotasbc  7331  setinds  9659  scottexs  9800  scott0s  9801  hta  9810  issubc  17791  dmdprd  19964  sbceqbidf  32544  bnj1454  34972  bnj110  34988  sbceqbii  36361  cbvsbcvw2  36400  cbvsbcdavw  36427  cbvsbcdavw2  36428  bj-csbsnlem  37198  rdgssun  37682  frege54cor1c  44330  frege55lem1c  44331  frege55c  44333