Definition df-sbc 3730

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3739). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3760 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 3731 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 3731, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3730 in the form of sbc8g 3737. 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 3730 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3738 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3731.

The related definition df-csb 3839 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 3729	. 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  3731  dfsbcq2  3732  sbceqbid  3736  sbcex  3739  nfsbc1d  3747  nfsbcdw  3750  nfsbcd  3753  sbc5  3757  sbc6g  3759  cbvsbcw  3762  cbvsbcvw  3763  cbvsbc  3764  sbcieg  3769  sbcied  3773  sbcbid  3784  sbcbi2  3788  sbcimdv  3798  sbcg  3802  intab  4921  brab1  5134  iotacl  6482  riotasbc  7339  setinds  9667  scottexs  9808  scott0s  9809  hta  9818  issubc  17799  dmdprd  19972  sbceqbidf  32553  bnj1454  34981  bnj110  34997  sbceqbii  36370  cbvsbcvw2  36409  cbvsbcdavw  36436  cbvsbcdavw2  36437  bj-csbsnlem  37207  rdgssun  37691  frege54cor1c  44339  frege55lem1c  44340  frege55c  44342