Definition df-sbc 3727

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3736). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3758 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 3728 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 3728, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3727 in the form of sbc8g 3734. 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 3727 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3735 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3728.

The related definition df-csb 3843 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 3726	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2714	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2105	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 205	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3728  dfsbcq2  3729  sbceqbid  3733  sbcex  3736  nfsbc1d  3744  nfsbcdw  3747  nfsbcd  3750  sbc5  3754  sbc6g  3756  cbvsbcw  3760  cbvsbcvw  3761  cbvsbc  3762  sbcieg  3766  sbcied  3771  sbcbid  3784  sbcbi2  3788  sbcimdv  3800  sbcg  3805  intab  4922  brab1  5135  iotacl  6451  riotasbc  7291  scottexs  9716  scott0s  9717  hta  9726  issubc  17620  dmdprd  19669  sbceqbidf  30944  bnj1454  32927  bnj110  32943  setinds  33853  bj-csbsnlem  35145  rdgssun  35605  frege54cor1c  41744  frege55lem1c  41745  frege55c  41747