Definition df-sbc 3743

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3752). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3773 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 3744 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 3744, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3743 in the form of sbc8g 3750. 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 3743 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3751 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3744.

The related definition df-csb 3852 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 3742	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2741	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2147	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 209	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is used by:  dfsbcq  3744  dfsbcq2  3745  sbceqbid  3749  sbcex  3752  nfsbc1d  3760  nfsbcdw  3763  nfsbcd  3766  sbc5  3770  sbc6g  3772  cbvsbcw  3775  cbvsbcvw  3776  cbvsbc  3777  sbcieg  3781  sbcied  3785  sbcbid  3796  sbcbi2  3800  sbcimdv  3810  sbcg  3814  intab  4941  brab1  5156  iotacl  6540  riotasbc  7415  setinds  9774  scottexsOLD  9967  scott0bsOLD  9969  hta  9986  htaOLD  9987  issubc  18036  dmdprd  20241  sbceqbidf  33152  bnj1454  35539  bnj110  35555  sbceqbii  37157  cbvsbcvw2  37196  cbvsbcdavw  37223  cbvsbcdavw2  37224  bj-csbsnlem  37992  rdgssun  38492  frege54cor1c  45105  frege55lem1c  45106  frege55c  45108