Definition df-sbc 3739

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3748). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3769 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 3740 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 3740, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3739 in the form of sbc8g 3746. 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 3739 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3747 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3740.

The related definition df-csb 3848 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 3738	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2712	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2113	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 206	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3740  dfsbcq2  3741  sbceqbid  3745  sbcex  3748  nfsbc1d  3756  nfsbcdw  3759  nfsbcd  3762  sbc5  3766  sbc6g  3768  cbvsbcw  3771  cbvsbcvw  3772  cbvsbc  3773  sbcieg  3778  sbcied  3782  sbcbid  3793  sbcbi2  3797  sbcimdv  3807  sbcg  3811  intab  4931  brab1  5144  iotacl  6476  riotasbc  7331  setinds  9656  scottexs  9797  scott0s  9798  hta  9807  issubc  17757  dmdprd  19927  sbceqbidf  32510  bnj1454  34947  bnj110  34963  sbceqbii  36334  cbvsbcvw2  36373  cbvsbcdavw  36400  cbvsbcdavw2  36401  bj-csbsnlem  37047  rdgssun  37522  frege54cor1c  44098  frege55lem1c  44099  frege55c  44101