Definition df-sbc 3713

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3722). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3744 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 3714 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 3714, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3713 in the form of sbc8g 3720. 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 3713 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3721 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3714.

The related definition df-csb 3830 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 3712	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2716	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2112	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 209	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3714  dfsbcq2  3715  sbceqbid  3719  sbcex  3722  nfsbc1d  3730  nfsbcdw  3733  nfsbcd  3736  sbc5  3740  sbc6g  3742  cbvsbcw  3746  cbvsbcvw  3747  cbvsbc  3748  sbcieg  3752  sbcied  3757  sbcbid  3770  sbcbi2  3775  sbcimdv  3787  sbcg  3792  intab  4906  brab1  5118  iotacl  6401  riotasbc  7228  scottexs  9551  scott0s  9552  hta  9561  issubc  17441  dmdprd  19491  sbceqbidf  30711  bnj1454  32697  bnj110  32713  setinds  33635  bj-csbsnlem  34990  rdgssun  35455  frege54cor1c  41385  frege55lem1c  41386  frege55c  41388