Definition df-sbc 3745

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3754). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3775 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 3746 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 3746, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3745 in the form of sbc8g 3752. 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 3745 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3753 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3746.

The related definition df-csb 3854 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 3744	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2707	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2109	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 206	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3746  dfsbcq2  3747  sbceqbid  3751  sbcex  3754  nfsbc1d  3762  nfsbcdw  3765  nfsbcd  3768  sbc5  3772  sbc6g  3774  cbvsbcw  3777  cbvsbcvw  3778  cbvsbc  3779  sbcieg  3784  sbcied  3788  sbcbid  3799  sbcbi2  3803  sbcimdv  3813  sbcg  3817  intab  4931  brab1  5143  iotacl  6472  riotasbc  7328  scottexs  9802  scott0s  9803  hta  9812  issubc  17760  dmdprd  19897  sbceqbidf  32449  bnj1454  34811  bnj110  34827  setinds  35754  sbceqbii  36167  cbvsbcvw2  36206  cbvsbcdavw  36233  cbvsbcdavw2  36234  bj-csbsnlem  36879  rdgssun  37354  frege54cor1c  43891  frege55lem1c  43892  frege55c  43894