Definition df-sbc 3743

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3752). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3773 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 3744 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 3744, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3743 in the form of sbc8g 3750. 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 3743 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3751 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3744.

The related definition df-csb 3852 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 3742	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2715	. . 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  3744  dfsbcq2  3745  sbceqbid  3749  sbcex  3752  nfsbc1d  3760  nfsbcdw  3763  nfsbcd  3766  sbc5  3770  sbc6g  3772  cbvsbcw  3775  cbvsbcvw  3776  cbvsbc  3777  sbcieg  3782  sbcied  3786  sbcbid  3797  sbcbi2  3801  sbcimdv  3811  sbcg  3815  intab  4935  brab1  5148  iotacl  6488  riotasbc  7345  setinds  9672  scottexs  9813  scott0s  9814  hta  9823  issubc  17773  dmdprd  19946  sbceqbidf  32579  bnj1454  35024  bnj110  35040  sbceqbii  36413  cbvsbcvw2  36452  cbvsbcdavw  36479  cbvsbcdavw2  36480  bj-csbsnlem  37178  rdgssun  37660  frege54cor1c  44300  frege55lem1c  44301  frege55c  44303