Definition df-sbc 3754

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3763). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3784 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 3755 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 3755, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3754 in the form of sbc8g 3761. 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 3754 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3762 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3755.

The related definition df-csb 3863 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 3753	. 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  3755  dfsbcq2  3756  sbceqbid  3760  sbcex  3763  nfsbc1d  3771  nfsbcdw  3774  nfsbcd  3777  sbc5  3781  sbc6g  3783  cbvsbcw  3786  cbvsbcvw  3787  cbvsbc  3788  sbcieg  3793  sbcied  3797  sbcbid  3808  sbcbi2  3812  sbcimdv  3822  sbcg  3826  intab  4942  brab1  5155  iotacl  6497  riotasbc  7362  scottexs  9840  scott0s  9841  hta  9850  issubc  17797  dmdprd  19930  sbceqbidf  32416  bnj1454  34832  bnj110  34848  setinds  35766  sbceqbii  36179  cbvsbcvw2  36218  cbvsbcdavw  36245  cbvsbcdavw2  36246  bj-csbsnlem  36891  rdgssun  37366  frege54cor1c  43904  frege55lem1c  43905  frege55c  43907