Definition df-sbc 3766

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3775). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3796 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 3767 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 3767, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3766 in the form of sbc8g 3773. 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 3766 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3774 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3767.

The related definition df-csb 3875 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 3765	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2713	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2108	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 206	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3767  dfsbcq2  3768  sbceqbid  3772  sbcex  3775  nfsbc1d  3783  nfsbcdw  3786  nfsbcd  3789  sbc5  3793  sbc6g  3795  cbvsbcw  3798  cbvsbcvw  3799  cbvsbc  3800  sbcieg  3805  sbcied  3809  sbcbid  3820  sbcbi2  3824  sbcimdv  3834  sbcg  3838  intab  4954  brab1  5167  iotacl  6516  riotasbc  7378  scottexs  9899  scott0s  9900  hta  9909  issubc  17846  dmdprd  19979  sbceqbidf  32414  bnj1454  34819  bnj110  34835  setinds  35742  sbceqbii  36155  cbvsbcvw2  36194  cbvsbcdavw  36221  cbvsbcdavw2  36222  bj-csbsnlem  36867  rdgssun  37342  frege54cor1c  43886  frege55lem1c  43887  frege55c  43889